6.1 Perpendicular and Angle Bisectors - Big Ideas Learning
6.1
TEXAS ESSENTIAL
KNOWLEDGE AND SKILLS
G.2.B
G.5.C
G.6.A
Perpendicular and Angle Bisectors
Essential Question
What conjectures can you make about a point
on the perpendicular bisector of a segment and a point on the bisector of
an angle?
Points on a Perpendicular Bisector
Work with a partner. Use dynamic geometry software.
a. Draw any segment
Sample
¡ª.
and label it AB
Points
A
3
Construct the
A(1, 3)
C
perpendicular
B(2, 1)
¡ª.
bisector of AB
2
C(2.95, 2.73)
Segments
b. Label a point C
AB = 2.24
that is on the
1
CA = ?
B
perpendicular
¡ª
CB = ?
bisector of AB
¡ª
0
Line
but is not on AB .
3
4
5
0
1
2
?x + 2y = 2.5
¡ª
¡ª
c. Draw CA and CB
and find their
lengths. Then move point C to other locations on the perpendicular bisector and
¡ª and CB
¡ª.
note the lengths of CA
d. Repeat parts (a)¨C(c) with other segments. Describe any relationship(s) you notice.
Points on an Angle Bisector
SELECTING TOOLS
To be proficient in math,
you need to visualize
the results of varying
assumptions, explore
consequences, and compare
predictions with data.
Work with a partner. Use dynamic geometry software.
a. Draw two rays ?
AB and ?
AC to form ¡ÏBAC. Construct the bisector of ¡ÏBAC.
b. Label a point D on the bisector of ¡ÏBAC.
c. Construct and find the lengths of the perpendicular segments from D to the sides
of ¡ÏBAC. Move point D along the angle bisector and note how the lengths change.
d. Repeat parts (a)¨C(c) with other angles. Describe any relationship(s) you notice.
Sample
4
E
3
B
2
D
A
1
C
F
0
0
1
2
3
4
5
6
Points
A(1, 1)
B(2, 2)
C(2, 1)
D(4, 2.24)
Rays
AB = ?x + y = 0
AC = y = 1
Line
?0.38x + 0.92y = 0.54
Communicate Your Answer
3. What conjectures can you make about a point on the perpendicular bisector
of a segment and a point on the bisector of an angle?
4. In Exploration 2, what is the distance from point D to ?
AB when the distance
from D to ?
AC is 5 units? Justify your answer.
Section 6.1
Perpendicular and Angle Bisectors
305
6.1
Lesson
What You Will Learn
Use perpendicular bisectors to find measures.
Use angle bisectors to find measures and distance relationships.
Core Vocabul
Vocabulary
larry
Write equations for perpendicular bisectors.
equidistant, p. 306
Using Perpendicular Bisectors
Previous
perpendicular bisector
angle bisector
In Section 3.4, you learned that a perpendicular
bisector of a line segment is the line that is
perpendicular to the segment at its midpoint.
C
A
A point is equidistant from two figures when the
point is the same distance from each figure.
STUDY TIP
A perpendicular bisector
can be a segment, a ray,
a line, or a plane.
B
P
¡ª.
??
CP is a ¡Í bisector of AB
Theorems
Theorem 6.1 Perpendicular Bisector Theorem
In a plane, if a point lies on the perpendicular
bisector of a segment, then it is equidistant
from the endpoints of the segment.
¡ª, then CA = CB.
If ??
CP is the ¡Í bisector of AB
C
A
B
P
Proof p. 306; Ex. 21, p. 352
Theorem 6.2 Converse of the Perpendicular Bisector Theorem
In a plane, if a point is equidistant from the
endpoints of a segment, then it lies on the
perpendicular bisector of the segment.
If DA = DB, then point D lies on
¡ª.
the ¡Í bisector of AB
C
A
B
P
Proof Ex. 32, p. 312
D
Perpendicular Bisector Theorem
¡ª.
?? is the perpendicular bisector of AB
Given CP
C
Prove CA = CB
A
P
B
¡ª, ??
Paragraph Proof Because ??
CP is the perpendicular bisector of AB
CP is
¡ª
¡ª
perpendicular to AB and point P is the midpoint of AB . By the definition of midpoint,
AP = BP, and by the definition of perpendicular lines, m¡ÏCPA = m¡ÏCPB = 90¡ã.
¡ª ? BP
¡ª, and by the definition
Then by the definition of segment congruence, AP
of angle congruence, ¡ÏCPA ? ¡ÏCPB. By the Reflexive Property of Congruence
¡ª ? CP
¡ª. So, ¡÷CPA ? ¡÷CPB by the SAS Congruence Theorem
(Theorem 2.1), CP
¡ª ? CB
¡ª because corresponding parts of congruent triangles are
(Theorem 5.5), and CA
congruent. So, CA = CB by the definition of segment congruence.
306
Chapter 6
Relationships Within Triangles
Using the Perpendicular Bisector Theorems
Find each measure.
R
a. RS
From the figure, ??
SQ is the perpendicular bisector
¡ª. By the Perpendicular Bisector Theorem, PS = RS.
of PR
S
Q
So, RS = PS = 6.8.
6.8
P
b. EG
¡ª, ??
Because EH = GH and ??
HF ¡Í EG
HF is the
¡ª
perpendicular bisector of EG by the Converse of the
Perpendicular Bisector Theorem. By the definition of
segment bisector, EG = 2GF.
F
E
9.5
24
G
24
So, EG = 2(9.5) = 19.
H
c. AD
¡ª.
From the figure, ??
BD is the perpendicular bisector of AC
AD = CD
5x = 3x + 14
x=7
Perpendicular Bisector Theorem
C
3x + 14
B
D
Substitute.
5x
Solve for x.
A
So, AD = 5x = 5(7) = 35.
Solving a Real-Life Problem
M
L
Is there enough information in the diagram to conclude that point N lies on the
¡ª?
perpendicular bisector of KM
SOLUTION
¡ª ? ML
¡ª. So, LN
¡ª is a segment bisector of KM
¡ª. You do not know
It is given that KL
¡ª
¡ª
whether LN is perpendicular to KM because it is not indicated in the diagram.
K
¡ª.
So, you cannot conclude that point N lies on the perpendicular bisector of KM
N
Monitoring Progress
Help in English and Spanish at
Use the diagram and the given information to find the
indicated measure.
¡ª, and YZ = 13.75.
1. ??
ZX is the perpendicular bisector of WY
Z
Find WZ.
¡ª, WZ = 4n ? 13,
2. ??
ZX is the perpendicular bisector of WY
and YZ = n + 17. Find YZ.
3. Find WX when WZ = 20.5, WY = 14.8, and YZ = 20.5.
W
Section 6.1
X
Perpendicular and Angle Bisectors
Y
307
Using Angle Bisectors
D
B
C
In Section 1.5, you learned that an angle bisector is a ray that divides an angle into two
congruent adjacent angles. You also know that the distance from a point to a line is the
? is
length of the perpendicular segment from the point to the line. So, in the figure, AD
¡ª
the bisector of ¡ÏBAC, and the distance from point D to ?
AB is DB, where DB ¡Í ?
AB.
Theorems
A
Theorem 6.3 Angle Bisector Theorem
B
If a point lies on the bisector of an angle, then it is
equidistant from the two sides of the angle.
¡ª ¡Í ?
¡ª ¡Í ?
If ?
AD bisects ¡ÏBAC and DB
AB and DC
AC,
then DB = DC.
D
A
C
Proof Ex. 33(a), p. 312
Theorem 6.4 Converse of the Angle Bisector Theorem
If a point is in the interior of an angle and is equidistant
from the two sides of the angle, then it lies on the
bisector of the angle.
¡ª ¡Í ?
¡ª ¡Í ?
If DB
AB and DC
AC and DB = DC,
then ?
AD bisects ¡ÏBAC.
B
D
A
C
Proof Ex. 33(b), p. 312
Using the Angle Bisector Theorems
Find each measure.
G
a. m¡ÏGFJ
7
¡ª ¡Í ?
¡ª ¡Í ?
Because JG
FG and JH
FH and JG = JH = 7,
?
FJ bisects ¡ÏGFH by the Converse of the Angle
Bisector Theorem.
J
F
42¡ã
7
So, m¡ÏGFJ = m¡ÏHFJ = 42¡ã.
H
b. RS
PS = RS
5x = 6x ? 5
5=x
Angle Bisector Theorem
S
5x
Substitute.
Solve for x.
6x ? 5
P
R
So, RS = 6x ? 5 = 6(5) ? 5 = 25.
Monitoring Progress
Q
Help in English and Spanish at
Use the diagram and the given information to find the indicated measure.
? bisects ¡ÏABC, and DC = 6.9. Find DA.
4. BD
? bisects ¡ÏABC, AD = 3z + 7, and
5. BD
CD = 2z + 11. Find CD.
A
D
B
6. Find m¡ÏABC when AD = 3.2, CD = 3.2, and
m¡ÏDBC = 39¡ã.
308
Chapter 6
Relationships Within Triangles
C
Solving a Real-Life Problem
A soccer goalie¡¯s position relative to the ball and goalposts forms congruent angles, as
shown. Will the goalie have to move farther to block a shot toward the right goalpost R
or the left goalpost L?
L
B
R
SOLUTION
The congruent angles tell you that the goalie is on the bisector of ¡ÏLBR. By the Angle
Bisector Theorem, the goalie is equidistant from ?
BR and ?
BL .
So, the goalie must move the same distance to block either shot.
Writing Equations for Perpendicular Bisectors
Writing an Equation for a Bisector
y
P
y = 3x ? 1
4
2
SOLUTION
M(1, 2)
Q
?2
2
Write an equation of the perpendicular bisector of the segment with endpoints
P(?2, 3) and Q(4, 1).
4
x
¡ª. By definition, the perpendicular bisector of PQ
¡ª is perpendicular to
Step 1 Graph PQ
¡ª
PQ at its midpoint.
¡ª.
Step 2 Find the midpoint M of PQ
?2 + 4 3 + 1
2 4
M ¡ª, ¡ª = M ¡ª, ¡ª = M(1, 2)
2
2
2 2
Step 3 Find the slope of the perpendicular bisector.
(
) ( )
1?3
?2
1
¡ª=¡ª
slope of PQ
= ¡ª = ?¡ª
4 ? (?2)
6
3
Because the slopes of perpendicular lines are negative reciprocals, the slope
of the perpendicular bisector is 3.
¡ª has slope 3 and passes through (1, 2).
Step 4 Write an equation. The bisector of PQ
y = mx + b
Use slope-intercept form.
2 = 3(1) + b
Substitute for m, x, and y.
?1 = b
Solve for b.
¡ª is y = 3x ? 1.
So, an equation of the perpendicular bisector of PQ
Monitoring Progress
Q
P
7. Do you have enough information to conclude that ?
QS bisects ¡ÏPQR? Explain.
R
S
Help in English and Spanish at
8. Write an equation of the perpendicular bisector of the segment with endpoints
(?1, ?5) and (3, ?1).
Section 6.1
Perpendicular and Angle Bisectors
309
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