Module 1
e Essential Question
How are points, lines, and segments used to model the real world?
What Will You Learn?
How much do you already know about each topic before starting this module?
KEY
V — I don’t know. (S5>p — I’ve heard of it.
73 — I know it!
Before•
After
ci
analyze axiomatic systems and identify types of geometry
analyze figures to identify points, lines, planes, and intersections
of lines and planes
find measures of line segments
apply the Distance Formula to find lengths of line segments
find points that partition directed line segments on number lines
find points that partition directed line segments on the
coordinate plane
find midpoints and bisect line segments
[JP Foldables Make this Foldable to help you organize your notes about geometric concepts.
Begin with four sheets of 11" x 17" paper.
1. Fold the four sheets of paper in half.
2. Cut along the top fold of the papers.
Staple along the side to form a book.
3. Cut the right sides of each paper to
create a tab for each lesson.
4. Label each tab with a lesson
number.
Module 1 • Tools of Geometry 1
What Vocabulary Will You Learn?
analytic geometry
• defined term
• midpoint
axiom
• definition
• plane
axiomatic system
• directed line segment
• point
betweenness of points
• distance
• postulate
bisect
• equidistant
• segment bisector
collinear
• fractional distance
• space
congruent
• intersection
• synthetic geometry
congruent segments
• line
• theorem
coplanar
• line segment
• undefined terms
Are You Ready?
Complete the Quick Review to see if you are ready to start this module.
Then complete the Quick Check.
Quick Review
Example 1
Graph and label the point Q(—3, 4) in the
coordinate plane.
Start at the origin. Because the
x-coordinate is negative,
move 3 units to the left. Then
move 4 units up because the
/-coordinate is positive.
Draw a dot and label it O.
Example 2
Evaluate the expression [—2 — (—7)]2 + (1 — 8)2.
Follow the order of operations.
[-2 - (—7)]2 + (1 - 8)
= 52 + (—7)2
= 25 + 49
= 74
Subtract in parentheses.
Evaluate exponents.
Add.
Quick Check
Graph and label each
point on the coordinate
plane.
1. W(-5, 2)
2. X(0, 4)
3. X(-3. -1)
4. Z(4, -2)
Evaluate each expression.
5. (4 - 2)2 + (7 - 3)2
6. (-5 - 3)2 + (3 - 4)2
7. [-1 - (—9)]2 + (5 - 3)2
8. [-3 - (—4)]2 + [-1 - (-6)]2
How did you do?
Which exercises did you answer correctly in the Quick Check?
2 Module 1 • Tools of Geometry
Explore Using a Game to Explore Axiomatic Systems
0 Online Activity Use a real-world situation to complete the Explore.
@ INQUIRY What are the characteristics of a
good set of rules?
Learn The Axiomatic System of Geometry
Geometry is an axiomatic system based on logical reasoning and
axioms.
The Axiomatic System of Geometry
An axiomatic:
derived.
system has a set of axioms from which theorems can be
undefined
terms
words, usually readily understood, that are not formally
explained by means of more basic words and concepts
definition
an explanation that assigns properties to a mathematical
object
defined term
a term that has a definition and can be explained using
undefined terms and/or defined terms
axiom or
postulate
a statement that is accepted as true without proof
theorem
a statement or conjecture that can be proven true using
undefined terms, definitions, and axioms
Undefined and defined terms are used to write definitions. Undefined
terms, defined terms, and definitions are used to create axioms.
Undefined terms, defined terms, definitions, and axioms are used to
prove theorems.
One real-world axiomatic system that is probably familiar to you is the
set of rules to a game. The rules are the axioms, and they are used to
evaluate the legality of each play.
Today’s Goals
• Apply axioms to draw
conclusions.
• Identify examples of
synthetic and analytic
geometry.
Today’s Vocabulary
axiomatic system
undefined terms
definition
defined term
axiom
postulate
theorem
synthetic geometry
analytic geometry
Math History Minute
Thales (c. 624-546 B.C.)
was a Greek
mathematician,
philosopher, and
astronomer, and is the
first known individual
attributed with a
mathematical discovery.
He inspired Euclid,
Plato, and Aristotle,
who considered him to
be the first philosopher
in the Greek tradition.
Lesson 1-1 • The Geometric System 3
I
Talk About It!
What conclusion
cannot be made from
the provided axioms?
Study Tip
Theorems Theorems,
or conclusions, made
from a set of axioms
must be true in every
situation. It takes only
one example that
contradicts the
conjecture to show
that a theorem or
conclusion is not true.
0 Example 1 Apply an Axiomatic System
ANIMALS In the fictional country of Rythoth, blue animals are from
the mountains, and red animals are from the valleys. These animals
are categorized into three distinct classes: mammals, birds, and
reptiles. Mammals are covered by hair or fur, birds are covered by
feathers, and reptiles are covered by scales.
Part A Categorize the animals.
Write the name of each animal in the corresponding categories
in the table.
Birthplace
Mammal
Mountains
Valleys
Klub
Prit
Zog
Part B Use axioms.
Use the previously given axioms and the table you filled in to
draw three conclusions about the species of animals shown.
• The Rorx is a mammal from the mountains of Rythoth.
• The Zog is a reptile from the valleys of Rythoth.
• The Prit is a bird from the valleys of Rythoth.
4 Module 1 • Tools of Geometry
Check
PLANETS The fictional galaxy of Yogul contains at least 20 planets
including Mothera, Sothera, and Kothera. An animal can live on any
planet in the Yogul galaxy that contains its biome. Lizards and
scorpions live in the desert. Frogs and monkeys live in tropical forests.
Bears and foxes can be found in the tundra. The biomes of each planet
are permanent and will not change over time.
Color
Biome
—
desert
tropical
forest
tundra
Use the axioms given to determine what conclusions can be made
about the planets of Yogul. Select all that apply.
A. Bears and foxes can live on Sothera.
B. Lizards and scorpions can only live on Mothera.
C. Only frogs and monkeys can survive on Kothera.
D. Bears and foxes can survive on Sothera at temperatures as low
as —20°F.
E. All animals can live on Kothera.
F. Scorpions and lizards can live on Mothera.
Learn Types of Geometry
There are several types of geometry that are built upon different sets
of postulates including synthetic geometry and analytic geometry.
Synthetic geometry is the study of
geometric figures without the use
of coordinates. Synthetic geometry
is sometimes called pure geometry
or Euclidean geometry.
Analytic geometry is the study of
geometry using a coordinate
system. Analytic geometry is
sometimes called coordinate
geometry or Cartesian geometry.
© Think About It!
What is an advantage of
using analytic geometry
instead of synthetic
geometry?
Q Go Online You can complete an Extra Example online.
Lesson 1-1 • The Geometric System 5
Example 2 Identify Types of Geometry
Classify each figure as illustrating synthetic geometry or analytic
geometry.
analytic geometry
synthetic geometry
Check
Classify each figure as illustrating synthetic geometry or analytic geometry.
y
X
D,
B
7
6 /
X
c
6 Module 1 • Tools of Geometry
Practice
Q Go Online You can complete your homework online.
Example 1
1. BASKETBALL The Badgers’ basketball team has 10 players. During practice, half
of the players wear red jerseys numbered 1-5, and the other half wear yellow
jerseys numbered 6-10. The yellow team wins the practice game 32-26.
• Kylie wears number 5 and scores 9 points.
• Kelsey’s team wins the game.
• Marie and Kylie are on opposing teams.
Use the axioms to make three conclusions about the game played.
2. PRINTING Rico’s T-shirt Company sells customized short sleeve T-shirts, long
sleeve T-shirts, and sweatshirts. Each type of shirt sells in multiples of 5. It costs
$25.00 for 5 short sleeve T-shirts, $30.00 for 5 long sleeve T-shirts, and $40.00
for 5 sweatshirts. Short sleeve and long sleeve T-shirts can be made in any color
except navy or black. Sweatshirts are only made in navy and black.
• Mercedes bought green shirts for $55.00.
• Quinn bought 10 navy sweatshirts.
• Rachel paid $30.00 for several red shirts.
• Hector bought black and yellow shirts for $65.00.
Use the axioms to make four conclusions about the shirts sold.
3. LANDSCAPING Tom owns a landscaping business. He charges $40 for a yard
cleanup, $50 to mow a lawn, and $75 to mulch a yard. On average, it takes Tom
25 minutes for a yard cleanup, 40 minutes to mow a lawn, and 2 hours to mulch a
yard. Tom’s clients are Mr. Hansen, Ms. Martinez, and Mrs. Johnson.
• Mr. Hansen paid $125 for lawn services this week.
• Tom spent more than an hour at Ms. Martinez’ house this week.
• Mrs. Johnson wrote Tom a check for $165 for the week.
• Tom made $405 from his three clients this week.
Use the axioms to make four conclusions about the landscaping that Tom did.
4. CUPCAKES Olivia’s Cupcake Shoppe sells small and
large cupcakes in three flavors.
• Niamh paid $3 for a cupcake with buttercream icing.
• Bethany bought a small vanilla cupcake.
• Mateo paid $3.50 for a cupcake with strawberry icing
and a chocolate cupcake.
Use the axioms to make two conclusions about the
cupcakes that were purchased.
Flavors
• Chocolate with vanilla icing •
Vanilla with strawberry icing
- Strawberry with buttercream icing -
Sizes
Small......$1.75 Large...... $3.00
cupcake shopPe
Lesson 1-1 • The Geometric System 7
Example 2
Classify each figure as illustrating synthetic geometry or analytic geometry.
B
Mixed Exercises
11. RESTAURANT Damon sells three types of salads at his restaurant: cobb, wedge,
and spinach. Each salad is served with 2 dinner rolls. The price of the cobb salad
is $7.99, the price of the wedge salad is $8.99, and the price of the spinach salad
is $5.99. Grilled chicken can be added to any salad for an additional $2.00.
• Malik spent $7.99 on a salad.
• Pedro and Deandra each spent $8.99 on their salads.
• Rafael ate a wedge salad.
• Drake did not add chicken to his salad.
Use the axioms to make a conclusion about the salads that are eaten.
12. CLASSROOM Mrs. Fields teaches high school geometry. Her classroom tools
include a compass, straightedge, pencil, and protractor. Does Mrs. Fields likely
teach analytic geometry or synthetic geometry? Explain your reasoning.
13. REASONING Theo is stuck on a problem on a test. The problem is asking him to
use a given formula to find the distance between two points on a graph. Is Theo
using analytic geometry or synthetic geometry? Explain your reasoning.
14. USE A SOURCE Survey a group of students in your classroom about favorite
colors. Write three axioms about the data you collected. Then use your axioms to
write a conclusion. Explain your reasoning.
8 Module 1 • Tools of Geometry
Atomic
Imagery/Getty
Images, C
Squared
Studios/Photodisc/Getty
Images,
Tewin
Kijthamrongworakul/Alamy
Stock Photo
15. STATE YOUR ASSUMPTION Sydney is an engineer. She is
using a blueprint for a project that is drawn on a grid, as
shown. Is Sydney likely using analytic geometry or synthetic
geometry? Explain any assumptions that you make.
16. Mr. Sail assigns a project where students identify shapes
that represent real-world objects. Is this an example of
analytic geometry or synthetic geometry? Explain your
reasoning.
17. CONSTRUCT ARGUMENTS Consider the following axiomatic system for bus routes.
• Each bus route lists the stops in the order at which they are visited by the bus.
• Each route visits at least four distinct stops.
• No route visits the same stop twice, except for the first stop, which is always the
same as the last stop.
• There is a stop called Downtown, which is visited by each route.
• Every stop other than Downtown is visited by at most two routes.
The city has stops at Downtown, King St, Maxwell Ave, Stadium District, State
St, Grace Blvd, and Charlotte Ave. Are the following three routes a model for the
axiomatic system? Justify your argument.
ROUTE 1: Downtown, King St, Stadium District, State St, Downtown
ROUTE 2: Stadium District, State St, Grace Blvd, Maxwell Ave, Downtown,
Stadium District
ROUTE 3: King St, Stadium District, Downtown, Maxwell Ave, Stadium District, King St
18. SHOPPING The Clothing Shop is having a sale. All clothes are 20% off, and all
accessories are 30% off.
• Jaisa bought two necklaces.
• Sheree bought a shirt and a purse.
Use the axioms to make one conclusion about Jaisa or Sheree’s purchases.
19. WRITE Write a comparison of the rules and plays of a game and the elements of
an axiomatic system. Then choose a game or sport for which you know the rules.
Explain a rule from the game or sport and a play from the game. Does the play
violate or fall within the rule? Explain.
Lesson 1-1 • The Geometric System 9
20. CREATE Given the following list of axioms, draw a model to properly represent
the information.
• There exist five points.
• Each line contains only these five points.
• There exist two lines.
• Each line contains at least two points.
21. WHICH ONE DOESN’T BELONG? Three-point geometry is a finite subset of
geometry with the following four axioms:
• There exists exactly three distinct points.
• Each pair of distinct points are on exactly one line.
• Not all the points are on the same line.
• Each pair of distinct lines intersect in at least one point.
Which of the following does not satisfy all the axioms of three-point geometry? Justify
your conclusion.
22. FIND THE ERROR Grant read the following axioms for a video game he is playing.
• There are four keys hidden on each level.
• Each level ends when the player collects the third key.
• The game has 10 levels.
From these axioms, Grant concluded:
• to complete the game, he will need to find 30 keys.
• there are 40 keys in the game.
• he can collect all 40 keys in the game.
Are Grant’s conclusions correct? Explain your reasoning.
23. WHICH ONE DOESN’T BELONG? Using your understanding of analytic and synthetic
geometry, which of the following figures does not belong? Justify your conclusion.
10 Module 1 • Tools of Geometry
Lesson 1-2
Points, Lines, and Planes
Learn Points, Lines, and Planes
In geometry, point, line, and plane are considered undefined terms
because they are usually readily understood and are not formally
explained by means of more basic words and concepts.
You are already familiar with the terms point, line, and plane from
algebra. You graphed on a coordinate plane and found ordered pairs
that represented points on lines. In geometry, these terms have a
similar meaning.
Undefined Terms
A point is a location. It has neither shape nor size.
A
Named by a capital letter
Example point A
Example plane K, plane BCD, plane CDB,
plane DCB, plane DBC, plane CBD,
plane BDC
Space is defined as a boundless three-dimensional set of all points.
Space can contain lines and planes.
Collinear points are points that
lie on the same line. Noncollinear
points do not lie on the same line.
Coplanar points are points that lie
in the same plane. Noncoplanar
points do not lie in the same plane.
Points A, B, and C are collinear.
Points P, O, and R are coplanar.
Today’s Goals
• Identify points, lines,
and planes.
• Identify intersections
of lines and planes.
Today’s Vocabulary
point
line
plane
space
collinear
coplanar
intersection
Talk About It!
Can three points be
both noncollinear and
noncoplanar? Justify
your argument.
Lesson 1-2 • Points, Lines, and Planes 11
Study Tip
Additional Planes
Although not drawn,
there is another plane
that contains point S
and point T. Because
points S, T, and V are
noncollinear, points S
and T are in plane STV.
Example 1 Name Lines and Planes
Use the figure to name each of the following,
a. a line containing point Q
The line can be named as line c, or any two of the three points on
the line can be used to name the line.
Write the additional names
for line c below.
:TR RT TQ QT RQ OR
I
b. a plane containing point S and point T
One plane that can be named is plane A You can also use the
letters of any three noncollinear points to name this plane.
Plane TRS and plane TQS can be used to name this plane.
Circle another correct name for plane A
(plane OST) plane STV plane QVS
plane VST
@ Example 2 Model Points, Lines, and Planes
STUDENT
DESK Name the
geometric terms
modeled by the
objects in the
picture.
The notebook models
plane JKL or NJK.
The edges of the
notebook model lines
JK, KL, and JN.
The quarter models
point M in space.
12 Module 1 • Tools of Geometry
Points N, L, and K are coplanar.
Points P, O, and R are collinear.
Q Go Online You can complete an Extra Example online.
McCraw-Hill Education
Explore Intersections of Three Planes
Q Online Activity Use a concrete model to complete the Explore.
X
@ INQUIRY What figures can be formed by the
intersection of three planes?
Learn Intersections of Lines and Planes
The intersection of two or more geometric figures is the set of points
they have in common. Two lines intersect in a point. Lines can
intersect planes, and planes can intersect each other.
Example 3 Draw Geometric Figures
Draw and label a figure to represent the relationship.
Off and ST intersect at U for Q(-3, -2), R(4,1), S(2, 3), and T(-1, -5)
on the coordinate plane. Point V is coplanar with these points but
Draw and label a figure to represent the relationship.
JK and LM intersect at P for J(-4, 3), K(6, -3), L(-4, -5), and
M(3, 3) on the coordinate plane. Point Q is coplanar with these
points, but not collinear with JK and LM.
o Go Online You can complete an Extra Example online.
Lesson 1-2 • Points, Lines, and Planes 13
Study Tip
Dimensions A point
has no dimension. A line
exists in one dimension.
However, a circle is
two-dimensional, and
a pyramid is three-
dimensional.
Study Tip
Three-Dimensional
Drawings Because it
is impossible to show
an entire plane in a
figure, edged shapes
with different shades of
color are used to
represent planes.
Example 4 Interpret Drawings
Refer to the figure.
a. How many planes
appear in this figure?
six: plane P, plane
CAG, plane GFA, plane
EFA, plane DEA, and
plane DCA
b. Name four points that
are collinear.
Points H, I, C, and Fare collinear.
c. Name the intersection of plane GAC and plane P.
Plane GAC intersects plane P in GC.
<-»
«—>
_
d. At what point do JI and DC intersect? Explain.
It does not appear that
these lines intersect. DC
lies in plane P, but only
point / of JI lies in
plane P.
Check
Refer to the figure. Name
three points that are collinear.
Points _L_, ?
, and JL_ are
collinear.
E
@ Example 5 Model Intersections
AVIATION A biplane has two
main wings that are stacked
one above the other. Struts
connect the wings and are used
for support. Flying wires run
diagonally from the main body
of the plane to the wings and
between the stacked wings.
Complete the statements regarding the geometric terms modeled
by the biplane.
Each wing models a plane.
The intersection of a strut and a wing models a point.
The crossing of two flying wires models a point.
Q Go Online You can complete an Extra Example online.
14 Module 1 • Tools of Geometry
McGraw-Hill
Education
Practice
Q Go Online You can complete your homework online.
Example 1
Refer to the figure for Exercises 1-7.
1. Name the lines that are only in plane Q.
2. How many planes are labeled in the figure?
3. Name the plane containing the lines m and t.
4. Name the intersection of lines m and t.
5. Name a point that is not coplanar with points A, B,
and C.
6. Are points F, M, G, and P coplanar? Explain.
7. Does line n intersect line q? Explain.
Example 2
Name the geometric terms modeled by each object or phrase.
8. one solar panel
9. a tabletop
10. bridge support beam
13.
14. a wall and the floor
15. the edge of a table
16. two connected walls
17. a blanket
18. a telephone pole
19. a computer screen
Lesson 1-2 • Points, Lines, and Planes 15
Example 3
USE TOOLS Draw and label a figure for each relationship.
20. Points X and Y lie on CD.
21. Two planes that do not intersect.
22. Line m intersects plane R at a single point.
23. Three lines intersect at point J but do not all lie in the same plane.
24. Points 4(2, 3), B(2, —3), C, and D are collinear, but A, B, C, D, and Fare not.
Example 4
Refer to the figure for Exercises 25-28.
25. How many planes are shown in the figure?
26. How many of the planes contain points F and E?
27. Name four points that are coplanar.
28. Are points A, B, and C coplanar? Explain.
Example 5
29. BUILDING The roof and exterior walls of a house
represent intersecting planes. Using the image, name all
the lines that are formed by the intersecting planes.
30. If the surface of a lake represents a plane, what geometric
term is represented by the intersection of a fishing line
and the lake’s surface?
31. ART Perspective drawing is a method that artists use to
create paintings and drawings of three-dimensional
objects. The artist first draws the horizon line and two
vanishing points along the horizon. Buildings or other objects are created by
drawing receding lines and vertical lines.
a. Where do the receding lines and horizon lines intersect?
b. Identify examples of planes within this picture.
16 Module 1 • Tools of Geometry
Mixed Exercises
USE TOOLS Draw and label a figure for each relationship.
32. LM and NP are coplanar but do not intersect.
33. FG and JK intersect at P(4, 3), where point F is at (-2, 5) and point J is at (7, 9).
34. Lines s and t intersect, and line v does not intersect either one.
40. Plane J contains line s.
41. YP lies in plane B and contains point C, but does not contain point H.
42. Lines q and f intersect at point Z in plane U.
Lesson 1-2 • Points, Lines, and Planes 17
43. Name the geometric term modeled by the object.
44. Name the geometric term modeled by a partially-opened folder.
45. CREATE Sketch three planes that intersect in a point.
46. ANALYZE Is it possible for two points on the surface of a prism to be neither
collinear nor coplanar? Justify your argument.
47. FIND THE ERROR Camille and Hiroshi are trying to determine the greatest number
of lines that can be drawn using any four noncollinear points. Is either correct?
Explain your reasoning.
Camille
Because there are four points, 4 • 3
or 12 lines can be drawn between
the points.
Hiroshi
'you can draw 3 + 2 +1 or
6 lines between the points.
48. PERSEVERE What is the greatest number of planes determined using any three of
the points A, B, C, and D if no three points are collinear?
49. WRITE A finite plane is a plane that has boundaries or does not extend
indefinitely. The sides of the cereal box shown are finite planes. Give a
real-life example of a finite plane. Is it possible to have a real-life object that is
an infinite plane? Explain your reasoning.
50. CREATE Sketch three planes that intersect in a line.
18 Module 1 • Tools of Geometry
Learn Betweenness of Points
A line segment is a measurable part of a line that consists of two
points, called endpoints, and all of the points between them. The two
endpoints are used to name the segment.
For any two real numbers a and b, there is a real number n between
a and b such that a < n < b. This relationship also applies to points on
a line and is called betweenness of points.
Key Concept • Betweenness of Points
Point C is between A and B if and only if A, B, and C are collinear and
AC + CB = AB.
Example 1 Find Measurements by Adding
Find the measure of XZ.
XZ is the measure of XZ. Point Y is between
X and Z. Find XZ by adding XY and YZ.
XY + YZ = XZ
11.3 +3.8 =XZ
Betweenness of points
Substitution
15.1 cm =XZ
Add.
Check
Find the measure of DF.
o Go Online You can complete an Extra Example online.
Explore Using Tools to Determine Betweenness
of Points
O Online Activity Use a pencil and ruler to complete the Explore.
X
INQUIRY How can a line segment be divided
into any number of line segments?
Today’s Goals
• Calculate measures
of line segments.
• Apply the definition
of congruent line
segments to find
missing values.
Today’s Vocabulary
line segment
betweenness of
points
congruent
congruent segments
Talk About It!
What is an example of
how the betweenness
of points can be
applied to the real
world?
Lesson 1-3 • Line Segments 19
Problem-Solving Tip
Draw a Diagram Draw a
diagram to help you see
and correctly interpret a
situation that has been
described in words.
Think About It!
How can you check your
solution for x?
Q Think About It!
Once you find BC, how
could you find AC
without evaluating
AC = 4x - 12?
Example 2 Find Measurements by Subtracting
Find the measure of QR.
Point O is between points P and R.
PQ + QR = PR
6| + QR = 13}
Betweenness of
points
Substitution
5
Subtract 6 g from each side and simplify.
QR = 71 ft
Check
Find the measure of PQ. Round your
|«-------yj g
answer to the nearest tenth, if necessary. •------- ♦-------------
.1
cm
H---------- - 24.3 cm
Example 3 Write and Solve Equations to Find
Measurements
Find the value of x and BC if B is between A and C, AC = 4x — 12,
AB = x, and BC = 2x 4- 3.
Step 1 Plot two points and label them
A and C. Connect the points.
Step 2 Plot point B between points A
and C.
h----------- 4x —12 —
x
2x4-3
•--- •---------
A
B
Step 3 Label segments AB, BC, and AC with their given measures.
Step 4 Use betweenness of points to write an equation and solve for;
AC = AB + BC
4x - 12 = x 4- 2x 4- 3
4x — 12 = 3x 4- 3
x-12 = 3
x — 15
Now find BC.
BC = 2x + 3
= 2(15) 4- 3
= 33
Betweenness of points
Substitution
Combine like terms.
Subtract 3xfrom each side. Simplify.
Add 12 to each side. Simplify.
Given
x = 15
o Go Online You can complete an Extra Example online.
20 Module 1 • Tools of Geometry
@ Apply Example 4 Use Betweenness of Points
SPACE NEEDLE Darrell is visiting the Space Needle in Seattle,
Washington. He knows that the total height of the Space Needle
is 605 feet. The distance from the ground to the observation deck
is 10 feet more than six times the distance from the observation
deck to the top of the Space Needle. Help Darrell find the distance
from the ground to the observation deck.
1 What is the task?
Describe the task in your own words. Then list any questions that you
may have. How can you find answers to your questions?
I need to find the distance from the ground to the observation deck.
How does the distance from the ground to the observation deck
compare to the total height of the Space Needle? I can express the
information that I am given as an equation, solve for any missing
information, and then use that information to find the answer.
2 How will you approach the task? What have you learned that
you can use to help you complete the task?
I will express the information that I am given into an equation that
represents the total height of the Space Needle. I have learned how to
convert written information into expressions, and I have learned how to
solve equations.
3 What is your solution?
Use your strategy to solve the problem.
What equation represents the distance from the
ground to the top of the Space Needle?
x + 6x + 10 - 605
What is the distance from the ground to the
observation deck?
520 ft
4 How can you know that your solution is reasonable?
Q) Write About It! Write an argument that can be used to defend your
solution.
520 feet seems reasonable for the distance from the ground to the
observation deck. The distance from the observation deck to the top of
the Space Needle is 85 feet. The combined heights are realistic
compared to the total height.
Study Tip
Congruent Segments
Use a consecutive
number of tick marks
for each new pair of
congruent segments in
a figure. The segments
with two tick marks are
congruent, and the
segments with three
tick marks are
congruent.
F
Study Tip
Equal vs. Congruent
Lengths are equal, and
segments are congruent.
It is correct to say that
AB = CD and AB = CD.
However, it is not correct
to say that ZB = CD or
that AB = CD.
Watch Out!
Check Your
Answer Sometimes
solutions will result in
negative segment
lengths. If this occurs,
review your work
carefully. Either an
error was made, or
there is no solution.
Learn Line Segment Congruence
If two geometric figures have exactly the same shape and size, then
they are congruent. Two segments that have the same measure are
congruent segments.
Key Concept • Congruent Segments
= is read is congruent to. Red slashes on the figure also indicate
congruence.
Segment AB is congruent to segment CD.
Congruent segments have the same measure.
Example 5 Write and Solve Equations by Using
Congruence
Find the value of x if Q is between P and R,
PQ = 6x + 20, QR = 2(x + 6), and PQ = OR.
Write the justifications in the correct order. You
may use a justification more than once.
Definition of congruence
Divide each side by 4.
Distributive Property
Simplify. Substitution
Subtract 2x from each side. Subtract 20 from each side.
PQ = OR
6x + 20 = 2(x + 6)
6x + 20 = 2x + 12
6x + 20 - 2x = 2x + 12 - 2x
4x + 20 = 12
4x + 20 — 20 = 12 — 20
Definition of congruence
Substitution
Distributive Property
Subtract 2x from each side.
Simplify.
Subtract 20 from each side.
Simplify.
Divide each side by 4.
Simplify.
Check
Find the value of x if U is between T and V, TU = 7x + 35,
UV = 4(x + 7), and TU = UV.
x= ?
o Go Online You may want to complete the construction activities for this lesson.
22 Module 1 • Tools of Geometry
Practice
O Go Online You can complete your homework online.
Examples 1 and 2
Find the measure of each segment.
1. PR
7. NO
1 in. 1-rin.
•-------------•------ - -------- •
OP
N
8. AC
4.9 cm 5.2 cm
•---------------•----------------•
ABC
3. JL
0.75 cm
•-------------
J
0.35 cm
• I •
K L
9. GH
F 9.7 mm G
Fl
•-------------------•--------•
H----------15 mm----------- H
Example 3
Find the value of the variable and YZ if Y is between X and Z.
10. XY = 11, YZ = 4c, XZ = 83
11. XY = 6b, YZ = 8b, XZ = 175
12. XY = 7a, YZ = 5a, XZ = 6a + 24
13. XY = 5.5, YZ = 2c, XZ = 8.9
14. XY = 5n, YZ = 2n, XZ = 91
15. XY = 4w, YZ = 6w, XZ = 12w - 8
16. XY = 11d, YZ = 9d — 2, XZ = 5d + 28
17. XY=4n + 3, YZ = 2n - 7, XZ = 20
18. XY = 3a - 4, YZ = 6a + 2, XZ = 5a + 22
19. XY = 3k - 2, YZ = 7k + 4, XZ = 4k + 38
20. XY = 4x, YZ = x, and XZ = 25
21. XY = 4x, YZ = 3x, and XZ = 42
22. XY = 12, YZ = 2x, and XZ = 28
23. XY = 2x + 1, YZ = 6x, and XZ = 81
Lesson 1-3 • Line Segments 23
Example 4
24. RAILROADS A straight railroad track is being built to connect two cities. The
measured distance of the track between the two cities is 160.5 miles. A mail stop
that is between the two cities and lies along the track is 28.5 miles from the first
city. How far is the mail stop from the second city?
25. CARPENTRY A carpenter has a piece of wood that is 78 inches long. He wants to
cut it so that one piece is five times as long as the other piece. What are the
lengths of the two pieces?
26. WALKING Marshall lives 2300 yards from school and 1500 yards from the pharmacy.
The school, pharmacy, and his home are all collinear, as shown in the figure.
k--------------- 2300 yards---------------- H
1500 yards
•---------------•------------------------------•
School
Pharmacy
Home
What is the distance from the pharmacy to the school?
27. COFFEE SHOP Chenoa wants to stop for coffee on her way to school. The distance
from Chenoa’s house to the coffee shop is 3 miles more than twice the distance
from the coffee shop to Chenoa’s school. The total distance from Chenoa’s house
to her school is 5 times the distance from the coffee shop to her school.
a. What is the distance from Chenoa’s house to the coffee shop? Write your
answer as a decimal, if necessary.
b. What assumptions did you make when solving this problem?
Example 5
Find the measure of each segment.
28. MO
29. WY
4.6 cm
31. QT
32. DE
|<............. 14.4 in.-------------H
•
I •
I • 1 I
■ •.
P Q R
S
T
2x + 7
4(x - 3)
•-----
|------•-----
|----- •
C
D
E
30. FG
I*---------- 16.8 cm
• I • I • I
F G H
33. UX
•--------------------♦---------
1----------•------------------- •
U
V
W
X
3x + 1
4x — 6
2x + 8
24 Module 1 • Tools of Geometry
Mixed Exercises
34. Find the length of UW if W is between U and V, UV = 16.8 centimeters, and
VW = 7.9 centimeters.
6x — 4
10 cm
•------------------- •------------- •
35. Find the value of x if /?S = 24 centimeters.
R
T
S
36. Find the length of LO if M is between L and O, LM = 7x — 9, MO = 14 inches, and
LO = 10x - 7.
37. Find the value of x if PO = RS, PO - 9x - 7, and RS = 29.
38. Find the measure of NL.
----------------------------5.8 cm----------------------------*
2.1 cm
•---------------------- •----------------------------------------- •
MN
L
39. PRECISION If point P is between A and M, write a true statement.
40. HIKING A hiking trail is 20 kilometers long. Park organizers want to build 5 rest
stops for hikers with one on each end of the trail and the other 3 spaced evenly
between. How much distance will separate successive rest stops?
41. RACE The map shows the route of a race. You are at Y, 6000 feet from the first
checkpoint A. The second checkpoint B is located at the midpoint between A and
the end of the race Z. The total race is 3.1 miles. How far apart are the two
checkpoints?
•----------•------------ •----------- •
Y
A
B
Z
42. FIELD TRIP The marching band at Jefferson High School is taking a field trip from
Lansing, Michigan, to Detroit, Michigan. The bus driver was told to stop 53 miles
into the trip. If the rest of the trip is 41 miles and the entire journey can be
represented by the expression 3x 4-16, find the value of x.
Lesson 1-3 • Line Segments 25
43. DISTANCE Madison lives between Anoa and Jamie as depicted on the
line segment. The distance between Anoa’s house and Madison’s
house is represented by 3x + 2 miles, the distance between Madison’s
house and Jamie’s house is represented by 3x + 4 miles, and the
distance between Anoa’s house and Jamie’s house is represented by
9x — 3 miles. Find the value of x. Then find the distance between Madison’s
house and Jamie’s house.
44. FIREFIGHTING A firefighter training course is taking place in a
high-rise building. The high-rise building where they practice
is 48 stories high. If the emergency happens on the top floor and
the firefighters have already gone 29 stories, how many stories do
they still need to go?
45. CAFE You are waiting at the end of a long straight line at Coffee Express. Your
friend Denzel is r -I-12 feet in front of you. Denzel is 2r + 4 feet away from the
front of the line. If Denzel is in the exact middle of the line, how many feet away
are you from the front of the line?
46. REASONING For AC, write and solve an equation to
find AS.
47. PERSEVERE Point K is between points J and L. If JK = x2 — 4x, KL — 3x — 2, and
JL = 28, find JK and KL.
48. ANALYZE Determine whether the statement If point M is between points C and D,
then CD is greater than either CM or MD is sometimes, always, or never true.
Justify your argument.
49. PERSEVERE Point C is located between points B and D. Also, BC = 5x + 7,
CD = 3y + 4, BD = 38, and SO = 2x + 8y. Find the values of x and y.
50. WRITE If point B is between points A and C, explain how you can find AC if you
know AB and SC. Explain how you can find SC if you know AB and AC.
51. CREATE Sketch line segment AC. Plot point B between A and C.
Use a ruler to find AC and AB. Then write and solve an equation to
find BC.
26 Module 1 • Tools of Geometry
Learn Distance on a Number Line
The distance between two points is the length of the segment
between the points. The coordinates of the points can be used to find
the length of the segment.
Because PQ is the same as OP, the order in which you name the
endpoints is not important when calculating distance.
Example 1 Find Distance on a Number Line
Use the number line.
AB C DE F
H M I ♦ I I M I ♦
-5-4-3-2-1 0 1 2 3 4 5
Find CF.
CF = |x2 - x^
= |5 - (-1)|
= 6
Check
Use the number line.
Distance Formula
x1 = —1 and x2 = 5
Simplify.
A
B C D E
H ♦ I I I ♦ I ♦ I I ♦ I I ♦ I
-6-5-4-3-2-1 012345678
Find AE.
A. -12
B. 2
C. 12
D. 13
o Go Online You can complete an Extra Example online.
Today’s Goals
• Find the length of a line
segment on a number
line.
• Find the distance
between two points on
the coordinate plane.
Today’s Vocabulary
distance
Think About It!
Why do you think the
Distance Formula uses
absolute value?
Think About It!
Compare and contrast
the length of CF and the
length of FC.
Lesson 1-4 • Distance 27
Example 2 Determine Segment Congruence
Determine whether CB and DF are congruent.
AB C
DE
F
« I » + 1...4-......—I I +
I
+ »
-5 -4 -3 -2 -1
0 1 2
3 4
5
The coordinates of C and B are —1 and —3. The coordinates of D and
F are 2 and 5. Find the length of each segment.
CB — \x2 — Xji
= |-3-(-1)|
= |-2|
Distance Formula
Substitute.
Subtract.
= 2
Simplify.
The length of CB is 2 units.
DF = |x2 — x,|
Distance Formula
= 15 — 21
Substitute.
Subtract.
= 3
Simplify.
The length of DF is 3 units.
Because CB DF, the segments are not congruent.
Watch Out!
Subtraction with
Negatives Remember
that subtracting a
negative number is like
adding a positive
number.
Check
Determine whether AC and BD are congruent.
A
B C D E
« I ♦ I I I ♦ I » l-l fl I ♦ I »
-6-5-4-3-2-1 012345678
The segments
? congruent.
28 Module 1 • Tools of Geometry
Learn Distance on the Coordinate Plane
The endpoints of a segment on the coordinate plane can be used to
find the length of that segment by using the Distance Formula.
Key Concept • Distance Formula on the Coordinate Plane
If P has coordinates (xr and O has
coordinates (x2, y2), then
y
PQ = ^(x2 - xi)2 + (y2 - /i)2-
O(X2./2)
P(X1,/1)
X
Distance Formula
Substitute x2 and y2.
Substitute x1 and yr
= V(-7)2 + (-10)2
Subtract.
= V49 +100
Simplify.
149
Example 3 Find Distance on the Coordinate Plane
Find the distance between J(4, 3) and K(—3, —7).
Let J(4, 3) be (xv yj and K(-3, -7) be (x2> y2).
JK = ^(x2-xi)2 + (y2-yi)2
= V(-3-x1)2 + (-7-y1)2
= 7(-3 - 4)2 + (—7 - 3)2
Simplify.
The distance between J and K is V149 or approximately 12.2 units,
o Go Online An alternate method is available for this example.
Check
Find the distance between A and B.
n y
•
9)
o
1— c
B 5,
0
A
z
?-(>-<1—20
3x
-4
-6
-8
4(--6
4)
o Go Online You can complete an Extra Example online.
Think About It!
Compare and contrast
the Distance Formula
on a number line with
the Distance Formula
on the coordinate
plane.
Watch Out!
Simplify Radicals Do
not forget to leave your
answer in simplest
radical form when
using the Distance
Formula or the
Pythagorean Theorem.
Lesson 1-4 • Distance 29
Q Think About It!
Does your answer
seem reasonable? Why
or why not?
@ Example 4 Calculate Distance
INCLINE Chelsea and Arnie are sitting in separate cars on the
Monongahela Incline. Chelsea is traveling up Mount Washington
and Amie is traveling down. When the two girls notice each other,
Chelsea has a horizontal distance of 212.0 feet from the lower
station and is at a height of 151.6 feet. Amie has a horizontal
distance of 435.3 feet from the lower station and is at a height of
311.3 feet. What is the distance between the two girls?
Step 1 Draw a diagram.
Draw a diagram to represent the
situation. Label the x-axis as the
“Horizontal Distance from Lower
Station (in feet).” Label they-axis
as the “Height (in feet).” Use a
scale of 50 on the x-axis and the
y-axis.
Step 2 Use the Distance Formula.
Horizontal Distance from
Lower Station (in feet)
(xv y,) = (212.0,151.6) and (x2, y2) = (435.3, 311.3)
= ^(x2-xi)2 + (>,2-yi)2
Distance Formula
= 7(435.3 - 212.0)2 + (311.3 - 151.6)2
Substitute.
= V223.32 + 159.72
Subtract.
= 749,862.89 + 25,504.09
Square each term.
= 775,366.98
Add.
« 274.5
Take the positive
square root.
Chelsea and Amie are approximately 274.5 feet apart.
Check
SNOWBOARDING Manuel wants to go snowboarding with his friend.
The closest ski and snowboard resort is approximately 20 miles west
and 50 miles north of his house. Manuel picks up his friend who lives
15 miles south and 10 miles east of Manuel’s house. How far away are
the two boys from the resort?
?
•__mi
Q Go Online You can complete an Extra Example online.
30 Module 1 • Tools of Geometry
Practice
O Go Online You can complete your homework online.
Example 1
Use the number line to find each measure.
J K L M N P
"♦I I ♦ I » I ♦ I I ♦ I ♦ I »
-7-6-5-4-3-2-1 0 1 2 3 4 5 6
1. JL
2. JK
3. KP
4. NP
5. JP
Use the number line to find each measure.
E
F
G
H
J
K L
■ i ♦ n ♦ i
i ♦
i
i
♦ i
i n
'
-6
-4 -2
0
2
4
6
8
10
7. JK
8. LK
6. LN
9. FG
10. JG
11. EH
12. LF
Use the number line to find each measure.
J K
L M N
<| I I I f I I I I » I I I | I f I
-6 -4 -2
0 2
4
6
8
10
13. LN
14. JL
Example 2
Determine whether the given segments are congruent. Write yes or no.
ABC
D
E F
>l I I I ♦ I I I I I I If I I I I » I I
-10-9-8-7-6-5-4-3-2-1 0123456789 10
15. AB and EF
16. BD and DF
17. AC and CD
18. AC and DE
19. BE and CF
20. CD and DF
Example 3
Find the distance between each pair of points.
Lesson 1-4 • Distance 31
24. A(2, 6), /V(5,10)
25. /?(3, 4), 7(7, 2)
26. X(—3, 8), Z(—5,1)
Example 4
27. SPIRALS Denise traces the spiral shown in the figure. The spiral begins
at the origin. What is the shortest distance between Denise’s starting
point and her ending point?
28. ZOOLOGY A tiny songbird called the blackpoll warbler
migrates each fall from North America. A tracking study
showed one bird flew from Vermont at map coordinates (63, 45) to
Venezuela at map coordinates (67,10) in three days. If each map
coordinate represents 75 kilometers, how far did the bird travel?
>
o
X
29. CONSTRUCT ARGUMENTS Mariah is
training for a sprint-distance triathlon.
She plans on cycling from her house to
the library, shown on the grid with a scale
in miles. If the cycling portion of the
triathlon is 12 miles, will Mariah have
cycled at least g of that distance during
her bike ride? Justify your argument.
32 Module 1 • Tools of Geometry
Paul Reeves Photography/Shutterstock
30. SPORTS The distance between each base on a
baseball infield is 90 feet. The third baseman
throws a ball from third base to point P. To the
nearest foot, how far did the player throw the ball?
Mixed Exercises
Find the distance between each pair of points. Round
to the nearest tenth, if necessary.
31. M(—4, 9),/V(—5, 3)
32. C(2, 4), 0(5, 7)
Plate
33. A(5,1), 8(3, 6)
34. 1/(4, 4), X(5, 8)
35. S(6, 4), 7(3, 2)
36. M(-1, 8), A/(—3, 3)
37. W(-8,1), Y(0, 6)
38. 8(3, -4), C(5, -5)
39. 8(6,11), 7(3, —7)
40. A(—3, 8) and 8(-1, 4)
41. M(4, -3) and N(-2,1)
42. X(—3, 5) and Y(4, 2)
43. Use the number line to determine whether S\/ and UX are congruent. Writeyes
or no.
S
T
U
V
W X
• i*iiiii + iiii + iiiiiiimiiii + ii+»
-15
-10
-5
0
5
10
15
Name the point(s) that satisfy the given condition.
44. two points on the x-axis that are 10 units from (1, 8)
45. two points on the y-axis that are 25 units from (-24, 3)
46. Refer to the figure. Are VT and SU congruent?
Lesson 1-4 • Distance 33
47. KNITTING Mei is knitting a scarf with diagonal stripes. Before she began, she laid
out the pattern on a coordinate grid where each unit represented 2 inches. On the
grid, the first stripe began at (2, 0) and ended at (5, 4). All the stripes are the same
length. How many inches long is each stripe on the scarf?
48. ART A terracotta bowl artifact has a triangular pattern around the top,
as shown. All the triangles are about the same size and can be
represented on a coordinate plane with vertices at points (0, 6.8),
(4.5, 6.8), and (2.25, 0). If each unit represents 1 centimeter, what is
the approximate perimeter of each triangle, to the nearest tenth
of a centimeter?
49. ANALYZE Consider rectangle QRST with QR = ST = 4 centimeters and
RS = QT = 2 centimeters. If point U is on OR such that QU = UR and point \/is
on RS such that RV = VS, then is QU congruent to RV? Justify your argument.
50. WRITE Explain how the Pythagorean Theorem and the Distance Formula are
related.
51. PERSEVERE Point P is located on the segment between point A(1, 4) and
point D(7,13). The distance from A to P is twice the distance from P to D.
What are the coordinates of point P?
52. CREATE Plot points Y and Z on a coordinate plane. Then use the Distance
Formula to find YZ.
53. PERSEVERE Suppose point A is located at (1, 3) on a coordinate plane. If AB is 10
and the x-coordinate of point B is 9, explain how to use the Distance Formula to
find the y-coordinate of point B.
54.WRITE Explain howto use the Distance Formula to find the distance between
points (a, b) and (c, d).
34 Module 1 • Tools of Geometry
Lesson 1-5
Locating Points on a Number Line
Explore Locating Points on a Number Line with
Fractional Distance
Q Online Activity Use dynamic geometry software to complete the
Explore.
X
@ INQUIRY What general method can you use to
locate a point some fraction of the distance from
one point to another point on a number line?
Today’s Goals
• Find a point on a
directed line segment on
a number line that is a
given fractional distance
from the initial point
• Find a point that
partitions a directed line
segment on a number
line in a given ratio.
Today’s Vocabulary
directed line segment
fractional distance
Learn Locating Points on a Number Line with
Fractional Distance
While a line segment has two endpoints, a directed line segment has
an initial endpoint and a terminal endpoint.
Using a directed line segment enables you to calculate the coordinate
of an intermediary point some fraction of the length of the segment, or
fractional distance, from the initial endpoint.
Key Concept • Locating a Point at Fractional Distances on a Number Line
Find the coordinate of a point that is of the distance from point C
to point D.
Step 1 Calculate the difference C
D
of the coordinates of
I I I I I I I I I I I •
Xi
Xo
point C and point D.
(*2 “ X!)
Step 2 Multiply the difference
by the given fraction.
The fractional distance is
given by^(x2 -x,).
Step 3 Add the fractional
distance to the
coordinate of the
initial point xr
The coordinate of point P is
given byx1 + §(x2 -x^.
*1 + f (x2 - X!)
c
- x I
I
I
D
j i i i i i i i-
• T I I I
*1
1
i i r
1 x2
f (*2 ~ *1>
c
3
D
1
1
1 A
--- 1 I I I
*1
'1
1
1
1
1
1 T
*2
The coordinate of a point on a line segment with endpoints x1 and x2 is
given by x1 + ^(x2 - x^, where is the fraction of the distance.
Watch Out!
Don’t Use Absolute
Value When finding
the distance from an
initial endpoint to a
terminal endpoint on a
directed line segment,
don’t use absolute
value. The difference
created by (x2 - x^ can
be positive or negative.
The sign of the
difference will indicate
the direction of the
directed line segment.
Talk About It!
In the Key Concept,
what phrase helped you
identify the initial
endpoint? What phrase
helped you identify the
terminal endpoint?
Lesson 1-5 • Locating Points on a Number Line 35
Think About It!
How would you check
your solution?
Q Think About It!
What would the
coordinate be if Julio
wanted to rest g of the
distance if he is going
from the library to his
house?
Example 1 Locate a Point at a Fractional Distance
Find B on AC that is of the
distance from A to C.
A
C
>1 I I I I I I I I I U
-5-4-3-2-1 0 1 2 3 4 5 6 7
Points is the initial endpoint, and point C is the terminal endpoint.
A
C
- * I I I I I I I I I I I * -
-5-4-3-2-1 0 1 2 3 4 5 6 7
Use the equation to calculate the coordinate of point B.
B = x1 + (x2 — X-])
Coordinate equation
= -5 + | (7 - (-5))
xy = -5, x2 = 7, and § = |
= —2
Simplify.
Point B is located at —2 on the number line.
A B
C
I If I I I I I I I I
-5-4-3-2-1 0 1 2 3 4 5 6 7
Check
—
3
Find X on BE that is 5 of the distance from B to E.
B
E
> I I I I ♦ I I I I I I I I I I l>
-5-4-3-2-1 0123456789 10
A. 2
B. 3
C. 5
D. 6
0 Example 2 Locate a Point at a Fractional Distance
in the Real World
BIKING Julio is biking from his house to the library. His house is
8 blocks west of the school, and the library is 4 blocks east of the
school. If he stops to rest of the distance from his house to the
library, at what point does he stop?
Julio’s house is the initial endpoint, located at -8, and the library is the
terminal endpoint, located at 4. The school is at 0.
w —I...+. 1—I—I—I—I—I—1....4
I—I—I ♦
I—I—
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
1
2 3 4
5
6
7
Use the equation to calculate the coordinate of Julio’s resting point.
B = x1 +
(x2 — x^
Coordinate equation
= -8 + 5 [4 - (-8)]
= -8, x2 = 4, and § = |
= —4
Simplify.
Q Go Online You can complete an Extra Example online.
36 Module 1 • Tools of Geometry
Check

5
DECORATING Taji is hanging a picture g of the distance from the floor
to the ceiling. If the distance between the floor and the ceiling is
12 feet, how high should he hang the picture?
Learn Locating Points on a Number Line with a
Given Ratio
You can calculate the coordinate of an intermediary point that
partitions the directed line segment into a given ratio.
If C has coordinate x1 and D has coordinate x2, then a point P that
partitions the line segment in a ratio of m:n is located at
nx1 4- mx2
coordinate m + n > where m =/= —n.
nxy + mx2
m + n
CP
D
>—H----
1------1----- 1
0 I-----
1----- 1----- 1----- 1---- h—0-
x-]
m: n
x2
Key Concept • Section Formula on a Number Line
Example 3 Locate a Point on a Number Line When
Given a Ratio
Find B on AC such that the ratio of AB to BC is 3:4.
-5 -4 -3 -2 -1
0
1
2
3
4
5
6
7
Use the Section Formula to determine the coordinate of point B.
nx* + mx2
B = m'
Section Formula
= 3^4^^ — y
m = 3, n = 4, x1 = -5, and x2 = 7
AB
C
I I I I > I I I I I
-5-4-3-2-1 0 1 2 3 4 5 6 7
So, B is located at y on the number line.
(continued on the next page)
Q Go Online
You may want to
complete the Concept
Check to check your
understanding.
Study Tip
Checking Solutions
When using the
Section Formula you
can check your
solution by converting
the given ratio into a
fraction. Use this
fraction and the
coordinate equation to
find the fractional
distance from your
initial endpoint to your
terminal endpoint. If
you don’t calculate the
same coordinate, you
have made an error.
Lesson 1-5 • Locating Points on a Number Line 37
Think About It!
How can you use
estimation to check
your answer?
Check
Find P on AF such that the ratio of AP to PF is 1:3.
A
F
♦I ♦ I I I I-H- Illi H I- I I I I ♦ I
-16 —14 -12 -10 -8 -6 -4-2 0 2
P is located at
L_ on the number line.
0 Example 4 Partition a Directed Line Segment
ROAD TRIP Jorge is traveling
2563 miles from New York
City to San Francisco by car.
He plans on stopping for gas
when the ratio of the distance
he has already traveled to the
distance he still has to travel is
2:5. How far has Jorge
traveled when he stops for
gas?
Use the Section Formula to determine how far Jorge will travel before
he stops for gas.
nx1 + mx2
B — m + n
Section Formula
5(0) + 2(2563)
2 + 5
= 732.3
m = 2, n = 5, x1 = 0, and x2 = 2563
When Jorge has traveled 732.3 miles from New York City, the ratio
of the distance he has traveled to the distance that he still has to
travel is 2:5.
Check
ERRANDS Eduardo travels 30 miles from his house to the bike shop.
When Eduardo goes to the bike shop, he always stops at a local pizza
place that is along the way. The ratio of the distance Eduardo travels
from his house to the pizza place to the distance he travels from the
pizza place to the bike shop is 2:3.
How far is the pizza place from Eduardo’s house?
Q Go Online You can complete an Extra Example online.
38 Module 1 • Tools of Geometry
Practice
O Go Online You can complete your homework online.
Examples 1 and 3
Refer to the number line.
M
J
I I I I I I I I I I I I I I I »-4-
23456789 10 11 12 13 141516 17 18 19
Refer to the number line.
ABC
DE F
«4 I 4 4 I I I I 4 4- IUII
-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7
1. Find the coordinate of point B that is of
the distance from M to J.
2
7. Find the coordinate of point G that is g of
the distance from B to D.
2. Find the coordinate of point C that is of
the distance from M to J.
8. Find the coordinate of point H that is of
the distance from C to F.
3. Find the coordinate of point D that is of
the distance from M to J.
9. Find the coordinate of point J that is g of the
distance from A to E.
4. Find the coordinate of point X such that the
ratio of MX to XJ is 3:1.
4
10. Find the coordinate of point K that is g of
the distance from A to F.
5. Find the coordinate of point X such that the
ratio of MX to XJ is 2:3.
11. Find the coordinate of point X such that the
ratio of AX to XF is 1:3.
6. Find the coordinate of point X such that the
ratio of MX to XJ is 1:1.
12. Find the coordinate of point X such that the
ratio of BX to XF is 3:2.
13. Find the coordinate of point X such that the
ratio of CX to XE is 1:1.
14. Find the coordinate of point X such that the
ratio of FX to XD is 5:3.
Lesson 1-5 • Locating Points on a Number Line 39
Refer to the number line.
ABC D F
« + I + I + I I » I ♦ l>
-5-4-3-2-1 0 1 2 3 4 5
15. Find the coordinate of point X on AF that is 4 of the distance from A to F.
16. Find the coordinate of point Y on AC that is of the distance from A to C.
Refer to the number line.
A W X
Y Z E
■ I I I + I -+ I I + I I I I + I I ■»" I + I I
-10-9-8-7-6-5-4-3-2-1 0123456789 10
---- 2
17. Which point on AE is of the distance from A to E?
18. Point X is what fractional distance from E to A?
Find the coordinate of point M on AE that is of the distance from A to E.
19.
Refer to the number line.
F G H
J K L
■+III + III + IIIII + I + II + I'
-15
-10
-5
0
5
20. The ratio of FX to XK is 1:1. Which point is located at X?
21. Find the coordinate of Q on FL such that the ratio of FQ to QL is 12:7.
Examples 2 and 4
22. TRAVEL Caroline is taking a road trip on I-70 in Kansas. She stops for gas at mile
marker 36. Her destination is at mile marker 353 in Topeka, but she decides to
stop at an attraction of the way after stopping for gas. At about which mile
marker did Caroline stop to visit the attraction?
40 Module 1 • Tools of Geometry
23. HIKING A hiking trail is 24 miles from start to finish. There are two rest areas
located along the trail.
a. The first rest area is located such that the ratio of the distance from the start of
the trail to the rest area and the distance from the rest area to the end of the
trail is 2:9. To the nearest hundredth of a mile, how far is the first rest area from
the starting point of the trail?
b. Kadisha claims that the distance she has walked and that the distance she has
left to walk has a ratio of 5:7. How many miles has Kadisha walked?
24. Melany wants to hang a canvas, which is 8 feet wide, on his wall. Where on the
canvas should Melany mark the location of the hangers if the canvas requires a
hanger every of its length, excluding the edges? Justify your answer.
25. MIGRATION Many American White Pelicans
migrate each year, with hundreds of them
stopping to rest in various locations along the
way. The ratio of the distance some flocks travel
from their summer home to one stopover to the
distance from the stopover to the winter home is
3:4. If the total distance that the pelicans migrate
is 1680 miles, how long is the distance from the
summer home to the stopover?
Lesson 1-5 • Locating Points on a Number Line 41
Mixed Exercises
26. Write an equation that can be used to find the coordinate of point K
2
that is 5 of the distance from O to /?.
O
R
*4-1. 4 I I I I I I ♦ I I I I I l>
-5-4-3-2-1 0123456789 10
27. SOCIAL MEDIA Tito is posting a photo and needs to resize it to fit. The photo’s
width should fill of the width of the page. On Tito’s screen, the total width of the
page is 3 inches. How wide should the photo be?
28. NEONATAL At birth, the ratio of a baby’s head length to the length of the rest of its
body is 1:3. If a baby’s total body length is 22 inches, how long is the baby’s head?
29. CREATE Draw a segment and label it AB. Using only a compass and a
straightedge, construct a segment CD such that CD = 5^ AB. Explain and then
justify your construction.
30. WRITE Naoki wants to center a canvas, which is 8 feet wide, on his bedroom wall,
which is 17 feet wide. Where on the wall should Naoki mark the location of the
nails, if the canvas requires nails every -g- of its length, excluding the edges?
Explain your solution process.
31. ANALYZE Determine whether the following statement is sometimes, always, or
never true. Justify your argument.
----- 2
If XY is on a number line and point W is of the distance from Xto Y, then the
coordinate of point IV is greater than the coordinate of point X.
32. PERSEVERE On a number line, point A is at 5, and point B is at -10.
Point C is on AB such that the ratio of AC to CB is 1:3. Find D
----
3
on BC that is g of the distance from B to C.
42 Module 1 • Tools of Geometry
Lesson 1-6
Locating Points on a Coordinate Plane
Explore Applying Fractional Distance
© Online Activity Use a real-world situation to complete the Explore.
@ INQUIRY How do we use fractional distances
in the real world?
Today’s Goals
• Find a point on a directed
line segment on the
coordinate plane that is a
given fractional distance
from the initial point.
• Find a point that partitions
a directed line segment
on the coordinate plane
in a given ratio.
Learn Locating Points on the Coordinate Plane
with Fractional Distance
You can find a point on a directed line segment that is a fractional
distance from an endpoint on the coordinate plane.
Key Concept • Locating a Point at a Fractional Distance on the
Coordinate Plane
The coordinates of a point on a line segment that is of the distance
from initial endpoint ^(x^y^ to terminal endpoint C(x2, y2) are given by
(x1 + ^(x2 - xp, + ^(y2 - y^j, where is the fraction of the distance
ifb^O.
Watch Out!
Determine the Initial
Endpoint Direction is
important when
determining a point
that is a fractional
distance on a directed
line segment. Identify
the initial endpoint you
move from and the
terminal endpoint you
move toward.
Example 1 Fractional Distances on the Coordinate Plane
----
3
Find C on AB that is 4 of the distance from
A to B.
Step 1 Identify the endpoints.
Identify the initial and terminal endpoints.
(*1> y,) = (-7, -5) and (x2, y2) = (6, 8)
Step 2 Find the x- and /-coordinates.
4
A
, 8)
/\
—I 5-'
9 o;
X
4
A
A
-8
^5)
Study Tip
Checking Coordinates
You can check that you
have computed the
coordinates of C
correctly by finding the
lengths of AC and AB.
If is not equal to
then you have made an
error.
Find the coordinates of C using the formula for fractional distance.
(X1 + §(x2 “ xi)’ -^1 + §(^2 “ yj)
Fractional Distance Formula
(—7 + ^-[6 — (—7)], —5 4- ^[8 — (—5)]}
Substitution
Point C is located at (2.75, 4.75).
o Go Online You can complete an Extra Example online.
Q Think About It!
What are the
coordinates of a
3
point that is of the
distance from B to A?
Lesson 1-6 • Locating Points on a Coordinate Plane 43
Talk About It!
How could you check
the coordinates of
point C?
Check
Coordinates of point P ?
Learn Locating Points on the Coordinate Plane with a
Given Ratio
The Section Formula can be used to locate a point that partitions a
directed line segment on the coordinate plane.
If A has coordinates (xr and C
has coordinates (x2, y2), then a
point B that partitions the line
segment in a ratio of m:n has
coordinates
/nx1 + mx2 nyy + my2\
& \ m + n ’ m + n /’
where m^n.
Key Concept • Section Formula on the Coordinate Plane
I
Example 2 Locate a Point on the Coordinate Plane
When Given a Ratio
Point C is located at ( —
44 Module 1 • Tools of Geometry
A.
(4, 8)
B. (2,3)
C. (1J)
D.
(0, -1)
Check
Find S on QR such that the ratio of OS to
SR is 2:1.
y
’/?
u
L)
J
L
I
Lt5^i-:
—IO
>X
-4
o /
0 Example 3 Partition a Directed Line Segment on the
Coordinate Plane
ZIP LINES Kendrick is riding a zip line. The zip line is 1800 meters
long and starts at a platform 600 meters above the ground. After he
jumps, someone takes a picture of his descent. When the picture is
taken, the ratio of the distance Kendrick has traveled to the distance
he has remaining is 1:2. The picture will show the horizontal distance
from 400 meters to 1200 meters from the base of the platform and
the vertical distance from ground level to a height of 500 meters.
Will Kendrick be in the frame of the picture?
II
To determine whether Kendrick is in the frame of the picture, first,
determine the horizontal distance x of the zip line. Then, use this
information to determine Kendrick’s location using the Section Formula.
Step 1 Determine the horizontal distance x of the zip line.
a2 + b2 - c2
6002 + x2 = 18002
x « 1697.1
Pythagorean Theorem
Substitute.
Solve.
The horizontal distance of the zip line is about 1697.1 meters.
(continued on the next page)
o Go Online You can complete an Extra Example online.
Lesson 1-6 • Locating Points on a Coordinate Plane 45
Step 2 Model the area captured by the photograph.
Step 3 Determine Kendrick’s location on the zip line.
Use the Section Formula to calculate Kendrick’s coordinates.
nx1 + mx2 ny3 + my2
m + n
Section Formula
f 2(0) 4-1(1697.1) 2(600) + 1(0)\
...
= ------ ^72-------’----- r+2----- / Substitute.
= (565.7, 400)
Simplify.
Kendrick is at (565.7, 400) when the picture is taken.
Step 4 Graph Kendrick’s location to determine whether he is in the
frame.
Yes. Kendrick is in the frame when the picture is taken.
Check
travel Andre is traveling
from Jeffersonville to
Springfield. He plans to
stop for a break when the
distance he has traveled
and the distance he has
left to travel have a ratio of
3:7. Where should Andre
stop for his break?
A. (13,12.5) B. (22,12.5)
C. (-3,6.5) D. (-12,6.5)
Q Go Online You can complete an Extra Example online.
46 Module 1 • Tools of Geometry
Practice
Q Go Online You can complete your homework online.
Example 1
Find the coordinates of point X on the coordinate plane for each situation.
1. Point X on AB is of the
distance from A to B.
p
>B 2, 9)
o
c /
b
f A
-J —6 -4/20
3x
—4
p
4(-
5)
-5, —
Q
2. Point X on PS is of the
distance from R to S.
y
3
R
4, z0
z
I
—Li-:
-
t-ix
2
S(2, -2
3. Point X on JK is of the
distance from J to K.
' ii ' 1
y
-*1
-rp
z
1
-1O
4x
p
'(5,
3P
A
Example 2
Refer to the coordinate grid.
4. Find point X on AB such that the ratio of AX to XB is 1:3.
5. Find point Y on CD such that the ratio of DY to YC is 2:1.
6. Find point Z on EF such that the ratio of EZ to ZF is 2:3.
Examples 1 and 2
Refer to the coordinate grid.
7. Find point C on AB that is of the distance from A to B.
----
5
8. Find point O on PS that is g of the distance from P to S.
9. Find point W on UV that is y of the distance from U to V.
-----
3
10. Find point D on AB that is of the distance from A to B.
11. Find point Z on PS such that the ratio of RZ to ZS is 1:3.
12. Find point G on AB such that the ratio of AG to GB is 3:2
13. Find point E on UV such that the ratio of UE to EV is 3:4.
Lesson 1-6 • Locating Points on a Coordinate Plane 47
15. CITY PLANNING The United States Capitol is located at (2, —4) on
a coordinate grid. The White House is located at (—10,16) on the same
coordinate grid. Find two points on the straight line between the
United States Capitol and the White House such that the ratio is 1:3.
Mixed Exercises
Refer to the coordinate grid.
-----
3
16. Find X on MN that is of the distance from M to N.
17. Find Y on MN such that the ratio of MY to YN is 1:3.
Point D is located on MV. The coordinates of D are (0, —
c y
0
A
N
J
Z
. 1I
Lr 1-:
>x
Z
q0
4
C
IJ
Julianne wants to find point F on WX such that the ratio of WF to FX is 2:3.
a. What error did Julianne make when solving this problem?
b. What are the correct coordinates of point F?
22. ANALYZE Is the point one-third of the distance from (x,,^)
to (x2, y2) sometimes, always, or never the point
(x. + x2 y.+ y2\
—3—, —3—J? Justify your argument.
23. WRITE Point P is located on the segment between point A(1,4)
and point D(7,13). The distance from A to P is twice the
distance from Pto D. Explain how to find the fractional
distance that P is from A to D. What are the coordinates of
point P?
24. PERSEVERE Point C(6, 9) is located on the segment between point A(4, 8) and point
B. Point C is of the distance from A to B. What are the coordinates of point B?
25. CREATE Draw a line on a coordinate plane. Label two points on the line Fand G.
Locate a third point on the line between points F and G and label this point H.
The point H on FG is what fractional distance from Fto G?
48 Module 1 • Tools of Geometry
Lesson 1-7
Midpoints and Bisectors
Explore Midpoints
0 Online Activity Use paper folding to complete the Explore.
@ INQUIRY What general formula can you use
to find the midpoint of a line segment?
Learn Midpoints on a Number Line
The midpoint of a segment is the point halfway between the
endpoints of the segment. A point is equidistant from other points if it
is the same distance from them. The midpoint separates the segment
into two segments with a ratio of 1:1. So, you can use the Section
Formula to derive the Midpoint Formula.
Key Concept • Midpoint on a Number Line
If AB has endpoints atx., and x2 on a number line, then the
X “I” X
midpoint M erf AB has coordinate M = ■1 9 2.
A
M
B
I I
I I I * I I I .. I... I ■ • -
*1
X, +x2
x2
2
Example 1 Find the Midpoint on a Number Line
What is the midpoint of XZ?
Today’s Goals
• Find the coordinate of
a midpoint on a number
line.
• Find the coordinates of
the midpoint or endpoint
of a line segment on the
coordinate plane.
• Find missing values
using the definition of a
segment bisector.
Today’s Vocabulary
midpoint
equidistant
bisect
segment bisector
Watch Out!
Ratios Remember that
1:1 refers to the ratio of
the distances, not to
the measures of the
segments.
-I I I i I I I I I I I I I U I I'
-6-5-4-3-2-1 01 23456789 10
x1 + x2
M = —2—
Midpoint Formula
8 + (-3)
=----- 5----- Substitution
5
= 2 or 2.5
Simplify.
The midpoint of XZ is 2.5.
o Go Online You can complete an Extra Example online.
Think About It!
Would your answer be
different if you reversed
the order of x1 and x2?
Lesson 1-7 • Midpoints and Bisectors 49
Q Think About It!
How else could Aponi
have located the
midpoint?
Check
What is the midpoint of AF?
A
F
< ♦ I I I I J I I fl I. +-♦-+
-6-5-4-3-2-1 012345678
0 Example 2 Midpoints in the Real World
SIGNS Aponi works at a
vintage clothing store. She
wants to hang a new sign so
it is centered above the
dressing-room doors. Given
that the dressing-room
doors have the same width,
find the point along the wall
that Aponi should hang the
new sign.
Midpoint Formula
Substitution
Simplify.
Aponi should hang the sign 10.5 feet from the left side of the wall.
Check
DISTANCE Jorge travels from his school on 38th Street to the library on
62nd Street. He stops halfway there to take a break. Where does Jorge
stop to rest?
School
Library
Jorge stops at
?____
Q Go Online You can complete an Extra Example online.
50 Module 1 • Tools of Geometry
Learn Midpoints on the Coordinate Plane
The Section Formula can be used to derive the Midpoint Formula for a
segment on the coordinate plane.
Because the midpoint separates the line segment into a ratio of 1:1,
substitute 1 for m and n into the formula.
Section Formula
Substitution
= (*i+x2 yi+y2)
Midpoint Formula
Key Concept • Midpoint Formula on the Coordinate Plane
If PQ has endpoints at P(xv y^ and Q(x2, y2) on the coordinate plane,
Example 3 Find the Midpoint on the Coordinate Plane
Find the coordinates of M, the midpoint of AB, forA(—2,1)
and 8(8, 3).
M =
+ x2 yi+y2)
= W-¥)
= (f'j) or<3- 2)
Midpoint Formula
Substitution
Simplify.
Check
Find the coordinates of B, the midpoint of AC, for>4(—3, —2) and
C(5,10).
o Go Online You can complete an Extra Example online.
Talk About It!
Would the coordinates
of the midpoint be
different if you use
point A as (x2,y2) and
point B as (xvyj?
Explain.
Lesson 1-7 • Midpoints and Bisectors 51
Watch Out!
Midpoint Formula The
Midpoint Formula only
uses addition and
division. Think of the
coordinates of the
midpoint as the
average of the
x-coordinates and the
average of the
y-coordinates of the
given endpoints.
Study Tip
Check for
Reasonableness
Always graph the given
information and the
calculated coordinates
of the midpoint to check
the reasonableness of
your answer.
Think About It!
How can you use the
graph to determine
whether your answer is
reasonable?
Example 4 Find Missing Coordinates
Find the coordinates of A if p(3, Q is the midpoint of AB and B has
coordinates (8, 3).
First, substitute the known information into the Midpoint Formula.
Let A be (x? and B be (x2, y2).
- - fX1 + X2 Yl+M
m = —2—’ —2— /
Midpoint Formula
(3, j) = ( “^2—»^2— )
Substitution
Next, write two equations to solve for x1 and yv
Equation for x1
Multiply each side by 2.
Solve.
Equation for/j
Multiply each side by 2.
Solve.
The coordinates of A are (—2, —2).
Plot the points on a coordinate plane to check your answer for
reasonableness.
y
r> o
L>\ O,
F^(3, O.S
o
X
A( -2 A
Check
Find the coordinates of O if /?(6, —1) is the midpoint of OS and S has
coordinates (12, 4).
o Go Online You can complete an Extra Example online.
x1 + 8
3 = -V"
6 = x1 + 8
—2 = x1
1 _ y1 + 3
2 — 2
i=y, + 3
-2=yi
52 Module 1 • Tools of Geometry
Learn Bisectors
Because the midpoint separates the segment into two congruent
segments, we can say that the midpoint bisects the segment. Any
segment, line, plane, or point that bisects a segment is called a
segment bisector.
Example 5 Find Missing Measures
Find the measure of RT if T is the midpoint of RQ.
Because T is the midpoint, RT — TQ. Use this equation to solve for x.
RT = TQ
Definition of midpoint
2x + 3 = 4x — 5
Substitution
3 = 2x - 5
Subtract 2x from each side.
Think About It!
8 = 2x
Add 5 to each side.
Is there a way to find the
length of TO without
4 = x
Divide each side by 2.
calculating when you
Substitute 4 forx in the equation for RT.
know the length of RT?
Why or why not?
RT=2x + 3
Equation for RT
= 2(4) + 3
Substitution
= 11
Simplify.
Check
Find the measure of /?S if S is the midpoint of RT.
R 7x—5 S 6x + 4 T
A. 56
B. 58
C. 112
D. 116
Q Go Online You can complete an Extra Example online.
Lesson 1-7 • Midpoints and Bisectors 53
Think About It!
What concept are we
using when we say that
AC = AB + BC?
o Go Online
You may want to
complete the
construction activities
for this lesson.
Example 6 Find the Total Length
Find the measure of AC if B is the midpoint of AC.
Because B is the midpoint, AB = BC. Use this equation to solve for x.
AB = BC
5x — 3 = 2x + 9
Definition of midpoint
Substitution
3x - 3 = 9
Subtract 2x from each side.
3x = 12
Add 3 to each side.
x = 4
Divide each side by 3.
The length of AC is equal to the sum of AB and BC. So, to find the
length of AC, substitute 4 for x in the expression 5x - 3 + 2x + 9.
AC — 5x — 3 + 2x + 9
= 5(4) - 3 + 2(4) + 9
= 20-3 + 8 + 9
= 34
The measure of AC is 34.
Length of AC
x = 4
Multiply.
Simplify.
Check
Find the measure of AC if B is the midpoint of AC. Round your answer
to the nearest tenth, if necessary.
o Go Online You can complete an Extra Example online.
54 Module 1 • Tools of Geometry
Practice
Q Go Online You can complete your homework online.
Example 1
Use the number line to find the coordinate of the midpoint of each segment.
J K L M N P
I I ♦ I ♦ I ♦ I I ♦ I ♦ l»
-7-6-5-4-3-2-1 0 1 2 3 4 5 6
1. KM
2. JP
3. LN
4. MP
5. LP
6. JN
Use the number line to find the coordinate of the midpoint of each segment.
E F G H J K L
■I ♦
I l»
I I ♦
I I ♦
I I ♦
I If
I
-6 -4
-2
0
2
4
6
8
10
7. FK
8. HK
9. EF
10. FG
11. JL
12. EL
USE TOOLS Use the number line to find the coordinate of the midpoint of each segment.
A B
C
D
E
■i m i ♦ i i i i i i i ♦ i i i ♦ i-
-6 -4 -2
0
2
4
6
8 10 12
13. DE
14. BC
15. BD
16. AD
Example 2
17. HOME IMPROVEMENT Callie wants to build a
fence halfway between her house and her
neighbor’s house. How far away from Callie’s
house should the fence be built?
Callie’s house
18. DINING Calvino’s home is located at the midpoint between Fast Pizza and Pizza
Now. Fast Pizza is a quarter mile away from Calvino’s home. How far away is Pizza
Now from Calvino’s home? How far apart are the two pizzerias?
Lesson 1-7 • Midpoints and Bisectors 55
Example 3
Find the coordinates of the midpoint of a segment with the given endpoints.
19. (5,11), (3,1)
20. (7, -5), (3, 3)
21. (-8, -11), (2, 5)
22. (7, 0), (2, 4)
23. (-5,1), (2, 6)
24. (-4, -7), (12, -6)
25. (2, 8), (8, 0)
26. (9, -3), (5,1)
27. (22, 4), (15, 7)
28. (12, 2), (7, 9)
29. (-15, 4), (2, -10)
30. (-2, 5), (3, -17)
31. (2.4,14), (6, 6.8)
32. (-11.2, -3.4), (-5.6, -7.8)
Example 4
Find the coordinates of the missing endpoint if B is the midpoint of AC.
33. C(—5, 4), B(—2, 5)
34. 21(1, 7), B(—3,1)
35. A(-4, 2), 8(6, -1)
37. A(4, -0.25), B(—4, 6.5)
36. C(—6, -2), B(—3, -5)
38. C(|,-6),s(f,4)
Examples 5 and 6
Suppose M is the midpoint of FG. Find each missing measure.
39. FM = 5y + 13, MG = 5 — 3y, FG = ?
40. FM = 3x - 4, MG = 5x - 26, FG = ?
41. FM = 8a+ 1, FG = 42, o = ?
42. MG = lx - 15, FG = 33, x = ?
43. FM = 3n + 1, MG = 6 - 2n, FG = ?
44. FM = 12x - 4, MG = 5x + 10, FG = ?
45. FM — 2k — 5, FG = 18, k = ?
46. FG = 14a + 1, FM = 14.5, a = ?
47. MG = 13x + 1, FG = 15, x = ?
48. FG = 11x - 15.6, MG = 10.9, x = ?
Mixed Exercises
Find the coordinates of the missing endpoint if P is the midpoint of NQ.
49. N(2, 0), P(5, 2)
50. A/(5, 4), P(6, 3)
51. Q(3, 9), P(-1, 5)
56 Module 1 • Tools of Geometry
52. Find the value ofy if M is the midpoint of LN.
9y-4
6y + 5
•--------------------------------•
L
M
-•
N
53. CAMPING Troop 175 is designing a new campground by first
mapping everything on a coordinate grid. They found locations for
the mess hall and their cabins. They want the bathrooms to
be halfway between these two places. What are the coordinates
of the location of the bathrooms?
54. GAME DESIGN A computer software
designer is creating a new video game. The
designer wants to create a secret passage
that is halfway between the castle and the
bridge. Where should the secret passage
be located?
55. SCAVENGER HUNT Pablo is going to ask
Bianca to prom by sending her on a scavenger
hunt. At the end of the scavenger hunt, Pablo
will be standing halfway between the gazebo
and the ice cream shop in town. Where should
Pablo stand?
56. WALKING Javier walks from his home at point K to the Internet cafe at point O. If
the school at point H/is exactly halfway between Javier's house and the Internet
cafe, how far does Javier walk?
Lesson 1-7 • Midpoints and Bisectors 57
57. SCHOOL LIFE Bryan is at the library doing a research paper. He
leaves the library at point A and walks to the soccer field for a game
at point C. The supermarket at point B is exactly halfway between
the library and the soccer field. After Bryan’s first soccer game, he
walks to the supermarket to buy a snack, and then he walks back to
the soccer field for his second game. Not including the time spent at
the soccer game, how far does Bryan walk?
C
(2x + 6) m
(5x - 3) m
A1
58. REASONING A drone flying over a field of corn identifies a dry area.
The coordinates of the vertices of the area are shown. To what
coordinates should the portable irrigation system be sent to water
the dry area? Explain your reasoning.
L2
B
O 200 400 600 800 1000
59. PERSEVERE Describe a method of finding the midpoint of a segment that has one
endpoint at (0, 0). Derive the midpoint formula, give an example using your
method, and explain why your method works.
60. WRITE Explain how the Midpoint Formula is a special case of the Section Formula.
61. CREATE Construct AC given AB if B is the midpoint of AC.
58 Module 1 • Tools of Geometry
Q Essential Question
How are points, lines, and segments used to model the real world?
Module Summary
The Geometric System
• An axiomatic system has a set of axioms from
which theorems can be derived.
• Synthetic geometry is the study of geometric
figures without the use of coordinates.
• Analytic geometry is the study of geometry using
a coordinate system.
Lessons 1-2 through 1-4
Points, Lines, Line Segments, and Planes
• The terms point, line, and plane are undefined
terms because they are readily understood and
are not formally explained by means of more
basic words and concepts.
• Collinear points are points that lie on the same
line. Coplanar points are points that lie in the
same plane.
• The intersection of two or more geometric
figures is the set of points they have in common.
• Point C is between A and B if and only if A, B,
and C are collinear and AC + CB = AB.
• Two segments that have the same measure are
congruent segments.
• The distance between two points on a number
line is the absolute value of their difference.
• The distance between two points on a
coordinate plane, (xr y^ and (x2, y2), is
7(x2 - x,)2 + (y2 - y)2.
Locating Points
• If C has coordinate x, and D has coordinate x2,
then a point P that partitions the line segment in
a ratio of m:n is located at
nx 4- mx
coordinate m + n -
• The coordinates of point B that is of the
distance from point A(xv y^ to point C(x2, y2) are
(x1 + ^(x2-Xi),yi + §(y2 -y,)).
Lesson 1-7
Midpoints and Bisectors
• If AB has endpoints atx1 and x2 on a number line,
then the midpoint M of~AB
x 4“ x
has coordinate M = 1 2 2
• A midpoint separates a segment into two
congruent parts, so it bisects the segment.
Study Organizer
Foldables
Use your Foldable to review this
module. Working with a partner
can be helpful. Ask for
clarification of concepts as
needed.
Module 1 Review • Tools of Geometry 59
Test Practice
1. MULTI-SELECT Select all real-world objects
that model a line. (Lesson 1-2)
A. electric tablet
B. pool stick
C. scoop of ice cream
D. light pole
E. emoji
2. MULTI-SELECT Use the figure to name all
planes containing point W. (Lesson 1-2)
A. plane WT
B. plane VWX
C. plane RYV
D. plane VWZ
E. plane RYX
3. OPEN RESPONSE What geometric figures do
the pages of the book represent? (Lesson 1-2)
4. MULTIPLE CHOICE Which sequence
identifies the correct order for completing
the construction to copy a line segment
using a compass and straightedge?
(Lesson 1-3)
W
M
X
f <—♦-
P
>
Y
£<—•-
P
Z
M
-•
N
A. X, Y, Z, W
B. W, Z, X, Y
C. W, Y, X, Z
D. Z, X, W, Y
60 Module 1 Review • Tools of Geometry
5. OPEN RESPONSE Find the value of x if Q is
between P and R, PQ-5x —10, QR = 3(x + 4),
and PQ = QR. (Lesson 1-3)
P
Q
R
•------------------ •---------------------•
5x - 10
3(x + 4)
6. OPEN RESPONSE On a straight highway, the
distance from Loretta’s house to a park is
43 miles. Her friend Jamal lives along this
same highway between Loretta’s house and
the park. The distance from Loretta’s house to
Jamal's house is 31 miles. How many miles is
it from Jamal's house to the park? (Lesson 1-3)
7. MULTIPLE CHOICE Find the distance
between the two points on a coordinate
plane. (Lesson 1-4)
>4(5,1) and B(-3, -3)
A. 4V5
B. 4V3
C. 2V2
D. 2V3
8. OPEN RESPONSE True or false: XY = WZ
(Lesson 1-4)
W
X
Z
Y
■ I + I—I—F-0-H—I—I—I—4—I—I 0 I
-6 -5 -4 -3 -2 -1 0
1 2 3 4 5 6 7 8
9. MULTIPLE CHOICE The coordinates of A and
B on a number line are —7 and 9. The
coordinates of C and D on a number line are
—4 and 12. Are AB and CD congruent? If
yes, what is the length of each segment?
(Lesson 1-4)
A. no
B. yes; 16
C. yes; -16
D. yes; 8
10. OPEN RESPONSE The coordinate of point
X on PQ that is | of the distance from P to Q
IS
. (Lesson 1-5)
P
Q
■ I » I I I I
I
I
I
I
I
I I » I
-6 -5 -4 -3 -2 -1
0 1 2 3 4 5 6 7 8
11. MULTIPLE CHOICE On a number line, point S
is located at —3 and point T is located at 9.
Where is point R located on ST if the ratio of
SR to RT is 3:4? (Lesson 1-5)
A. T1
7
B. 2I
C. 4
D.
15
7
Module 1 Review • Tools of Geometry 61
12. MULTIPLE CHOICE Find point R on ST such
that the ratio of SR to RT is 1:2. (Lesson 1-6)
S(-6, 8
y
7(3, 2)’
O
X
A.
/?(—5, 6)
B.
/?(—3, 6)
C.
/?(—1.5, 5)
D. R(0, 4)
13. OPEN RESPONSE Alonso plans to go to the
animal shelter to adopt a dog and then take
the dog to Precious Pup Grooming Services.
The shelter is located at (-1, 9) on the
coordinate plane, while Precious Pup
Grooming Services is located at (11, 0) on
the coordinate plane. Find the location of
Alonso’s home if it is of the distance from
the shelter to Precious Pup Grooming
Services. (Lesson 1-6)
14. OPEN RESPONSE Find the coordinates of A if
/W(6, -1) is the midpoint of AB, and B has the
coordinates (8, —7). (Lesson 1-7)
15. MULTIPLE CHOICE Find the measure of YZ if
Y is the midpoint of XZ. (Lesson 1-7)
A. 2
B. 10
C. 16
D. 20
16. MULTIPLE CHOICE Find the /-coordinate
of the point M, the midpoint of AB, for
A(—3, 3) and 8(5, 7). (Lesson 1-7)
A. -1
B.
1
C. 2
D. 5
17. MULTIPLE CHOICE Points A and B are plotted
on a number line. What is the location of M,
the midpoint of AB, for A at —9 and 8 at 28?
(Lesson 1-7)
A. M is located at 18.5 on the number line.
B. M is located at 14 on the number line.
C. M is located at 9.5 on the number line.
10
D. M is located at y on the number line.
62 Module 1 Review • Tools of Geometry