Name Class 3-3
Date Perpendicular Lines
Going Deeper
Essential question: How can you construct perpendicular lines and prove theorems
about perpendicular bisectors?
Perpendicular lines are lines that intersect at right angles.
In the figure, line ℓ is perpendicular to line m and you write
ℓ ⊥ m. The right angle mark in the figure indicates that the
lines are perpendicular.
m
The perpendicular bisector of a line segment is a line
perpendicular to the segment at the segment’s midpoint.
1
⊥m
MCC9–12.G.CO.12
EXAMPLE
Video Tutor
Constructing a Perpendicular Bisector
___
Construct the perpendicular bisector of AB​
​  . Work directly on the figure below.
A
Place the point of your compass at A.
Using a compass setting
that is greater
___
than half the length of AB​
​  , draw an arc.
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A
B
B
Without adjusting the compass,
A
B
C
place the point of the compass
at B and draw an arc intersecting
the first arc at C and D.
‹___›
Use
a straightedge to draw CD​
​    . 
___
‹
›
CD​
​     is the perpendicular
___
bisector of ​AB​ .
C
A
C
B
A
D
B
D
REFLECT
1a. How can you use a ruler and protractor to check the construction?
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Lesson 3
You can use reflections and their properties to prove a theorem about perpendicular
bisectors. Refer to the diagram in the Proof below as you read this definition of reflection.
A reflection across line m maps a point A to its image B as follows.
___
• Line m is the perpendicular bisector of AB​
​  if and only if A is not on line m.
• The image of P is P if and only if P is on line m.
The notation ​r​m​(A) = B means that the image of point A after a reflection across line m is
point B. The notation ​rm​ ​(P) = P means that the image of point P is point P, which implies
that P is on line m.
2
MCC9–12.G.CO.9
PROOF
Perpendicular Bisector Theorem
m
If a point is on the perpendicular bisector of a segment, then it is
equidistant from the endpoints of the segment.
P
___
Given: P is on the perpendicular bisector m of ​AB​ .
Prove: PA = PB
A
B
Complete the following proof.
Consider the reflection across line m. Then ​rm​ ​(P) = P because
Also, ​rm​ ​(A) = B by the definition of reflection.
Therefore, PA = PB because
REFLECT
2b. What conclusion can you make about △KLJ in the figure? Explain.
2c. Describe the point on the perpendicular bisector of a segment that is
closest to the endpoints of the segment.
Module 3
60
K
M
L
J
Lesson 3
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2a. Suppose you use a compass
and straightedge to construct the perpendicular
___
bisector of a segment, ​AB​ . If you choose a point P on the perpendicular bisector,
how can you use your compass to check that P is equidistant from A and B?
The converse of the Perpendicular Bisector Theorem is
also true. In order to prove the converse, you will use
the Pythagorean Theorem.
a
Recall that the Pythagorean Theorem states that in a right
triangle with legs of length a and b and hypotenuse of
length c, ​a2​ ​+ ​b2​ ​= ​c2​ ​.
c
a2 + b2 = c2
b
MCC9–12.G.CO.9
3
PROOF
Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then it lies on the
perpendicular bisector of the segment.
Given: PA = PB
___
Prove: P is on the perpendicular bisector m of AB​
​  .
A Use the method of indirect proof. Assume the opposite of what
m
you want to prove and show this leads to a contradiction.
Assume
that point P is not on the perpendicular bisector m of ​
___
AB​ . Then when you draw a perpendicular from P to the line
containing A and B, the perpendicular___
intersects this line at a
point Q, which is not the midpoint of AB​
​  .
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B
P
A
Q
B
Complete the following to show that this assumption leads to a contradiction.
___
PQ​
​  forms two right triangles, △AQP and △BQP.
​AQ​2​+ ​QP​2​= ​PA​2​and ​BQ​2​+ ​QP​2​= ​PB​2​by
Subtract these equations: ​AQ​2​+ ​QP​2​= ​PA​2​
​BQ​2​+ ​QP​2​= ​PB​2​
​AQ​2​- ​BQ​2​= ​PA​2​- ​PB​2​
However, ​PA​2​- ​PB​2​= 0 because
Therefore, ​AQ​2​- ​BQ​2​= 0. This means
​AQ​2​= ​BQ​2​and AQ = BQ. This contradicts the
___
fact that Q is not the midpoint of AB​
​  . Thus, the initial
assumption must be incorrect,
___
and P must lie on the perpendicular bisector of AB​
​  .
REFLECT
3a. In the proof, once you know ​AQ​2​= ​BQ​2​, why can you conclude AQ = BQ?
3b. Explain how the converse of the Perpendicular Bisector Theorem justifies the
compass-and-straightedge construction of the perpendicular bisector of a segment.
Module 3
61
Lesson 3
The perpendicular bisector construction can be used as part of the method for drawing
the perpendicular to a line through a given point not on the line.
MCC9–12.G.CO.12
4
EXAMPLE
Constructing a Perpendicular to a Line
Construct a line perpendicular to line m that passes
through point P. Work directly on the figure at right.
A
P
Place the point of your compass at P.
Draw an arc that intersects line m at
two points, A and B.
m
P
A
B
B
m
___
Construct the perpendicular bisector of AB​
​  .
This line will pass through P and be perpendicular
to line m.
P
A
B
m
REFLECT
pra c t i c e
1.Construct the perpendicular bisector of the
segment shown below.
2.Construct a line perpendicular to line m that
passes through point P.
B
m
A
P
Module 3
62
Lesson 3
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4a. Does the construction still work if point P is on line m? Why or why not?
Name
Class
3-3
Date
Additional Practice
1. Construct a line perpendicular to line r.
2. Construct the perpendicular bisector
of CD .
r
C
D
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3. Construct a line perpendicular to m through P. Then, using your two
perpendicular lines, construct a right triangle that has P as a vertex and
a hypotenuse with length XY.
Y
X
Use the diagram to find the given quantity.
4.
5.
C
F
8 cm
A
6 cm
B
D
1.5 ft
G
E
GE = ________________________________
CB = ________________________________
Module 3
2 ft
2 ft
63
Lesson 3
Problem Solving
1. Use geometric constructions to find a single point that is equidistant
from A and B, and also equidistant from C and D. (Note: The distance
from the point to A does not have to be the same as the distance from
the point to C. It only matters that the point is equidistant from each
pair.)
A
D
B
C
2. If the two segments from Exercise 1 were arranged so that they were both part of the same
line, as shown below, could you still find a point that is equidistant from A and B, and also
equidistant from C and D? Explain why or why not.
A
B
D
C
_________________________________________________________________________________________
_________________________________________________________________________________________
Select the best answer.
3. The road from Westtown to Easttown is
the perpendicular bisector of the road
from Northtown to Southtown. Given
that fact and the distances marked on
the map, how far is Westtown from
Easttown?
Northtown
5 mi
Westtown
A 3 mi
Easttown
3 mi
B 4 mi
C 5 mi
Southtown
D 6 mi
Module 3
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Lesson 3
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