5.5: Proofs Using Perpendicular Bisectors
Brinkman
SKILLS TO AQUIRE
 Find lengths using properties of perpendicular bisectors.
 Write proof arguments using properties of reflections.
 Use the Perpendicular Bisector Theorem in real situations.
B
A
REVIEW Proof:
Given: ⨀𝑂 ≅ ⨀𝑃
𝑂𝐴 = 𝑃𝐶
R
C
O
P
Proof: 𝑂𝐵 = 𝑃𝑅
Conclusion
Justification
1. ⨀𝑶 ≅ ⨀𝑷
Given
2. 𝑶𝑨 = 𝑷𝑪
Given
3. 𝑶𝑨 = 𝑶𝑩
Def’n of Circle (circle O)
4. 𝑷𝑪 = 𝑷𝑹
Def’n of Circle (circle P)
5. 𝑶𝑩 = 𝑷𝑹
Transitive Prop. of = (steps 2 and 4)
Definition: Perpendicular Bisector Theorem: If a point is on the PERPENDICULAR BISECTOR
of a segment, then it is EQUIDISTASNT from the endpoints of the segment.
P
Given: m is the perpendicular bisector of 𝑋𝑌
n is the perpendicular bisector of 𝑌𝑍
Prove: X,Y, and Z lie on ⨀𝑂 with radius OX.
Y
X
O
Conclusion
n
Justification
Z
m
1. m is the perp. bisector of 𝑿𝒀
Given
2. n is the perp. bisector of 𝒀𝒁
Given
3. OX = OY
Perpendicular Bisector Thrm (line m)
4. OY = OZ
Perpendicular Bisector Thrm (line n)
5. OX = OZ
Transitivity Property of =
6. X, Y, Z on ⨀𝑶, radius OX.
Definition of a Circle (Since OX, OY, OZ are radii, and the
center is O, then X, Y and Z, must be on circle O.
Practice
1.) m is the ┴ bisector of AB at the picture at the right.
A
a.) Name 2 pairs of congruent segments.
C
𝑨𝑪 and 𝑨𝑩
𝑨𝑷 and 𝑩𝑷
b.) If AP = 10, then BP = 10.
P
m
B
c.) If AB = 10, then AC = 5
2.) A tree stands midway between two states. Guy wires are attached from the states to a narrow
collar around the trunk. Explain why the guy wires must have the same length. Use picture to better
understand.
Look at the picture:
The tree is acting as a perpendicular bisector
to the ground. Since the guy wires are attached
to the tree at the same spot, and since the tree
is the midway point between the two stakes,
they must be the same distance to the ground.