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Perpendicular Bisectors

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What if you were given \triangle FGH and told that  \overleftrightarrow{GJ} was the perpendicular bisector of \overline{FH} ? How could you find the length of FG given the length of GH ? After completing this Concept, you'll be able to use the Perpendicular Bisector Theorem to solve problems like this one.

Watch This

CK-12 Perpendicular Bisectors

First watch this video.

James Sousa: Constructing Perpendicular Bisectors

Next watch this video.

James Sousa: Proof of the Perpendicular Bisector Theorem

Then watch this video.

James Sousa: Proof of the Perpendicular Bisector Theorem Converse

Finally, watch this video.

James Sousa: Determining Values Using Perpendicular Bisectors

Guidance

A perpendicular bisector is a line that intersects a line segment at its midpoint and is perpendicular to that line segment, as shown in the construction below.

One important property related to perpendicular bisectors is that if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. This is called the Perpendicular Bisector Theorem.

If \overleftrightarrow{CD} \perp \overline{AB} and AD = DB , then AC = CB .

In addition to the Perpendicular Bisector Theorem, the converse is also true.

Perpendicular Bisector Theorem Converse: If a point is equidistant from the endpoints of a segment, then the point is on the perpendicular bisector of the segment.

Using the picture above: If AC = CB , then \overleftrightarrow{CD} \perp \overline{AB} and AD = DB .

When we construct perpendicular bisectors for the sides of a triangle, they meet in one point. This point is called the circumcenter of the triangle.

Example A

If \overleftrightarrow{MO} is the perpendicular bisector of \overline{LN} and LO = 8 , what is ON ?

By the Perpendicular Bisector Theorem, LO = ON . So, ON = 8 .

Example B

Find x and the length of each segment.

\overleftrightarrow{WX} is the perpendicular bisector of \overline{XZ} and from the Perpendicular Bisector Theorem WZ = WY .

2x + 11 &= 4x - 5\\16 &= 2x\\8 &= x

WZ = WY = 2(8) + 11 = 16 + 11 = 27 .

Example C

Find the value of x . m is the perpendicular bisector of AB .

By the Perpendicular Bisector Theorem, both segments are equal. Set up and solve an equation.

3x-8 &=2x\\x &=8

CK-12 Perpendicular Bisectors

Guided Practice

1. \overleftrightarrow{OQ} is the perpendicular bisector of \overline{MP} .

a) Which line segments are equal?

b) Find x .

c) Is L on \overleftrightarrow{OQ} ? How do you know?

2. Find the value of x . m is the perpendicular bisector of AB .

3. Determine if \overleftrightarrow{S T} is the perpendicular bisector of \overline{XY} . Explain why or why not.

Answers:

1. a) ML = LP, \ MO = OP , and MQ = QP .

b) 4x + 3 & = 11\\4x & = 8\\x & = 2

c) Yes, L is on \overleftrightarrow{OQ} because ML = LP (the Perpendicular Bisector Theorem Converse).

2. By the Perpendicular Bisector Theorem, both segments are equal. Set up and solve an equation.

x+6 &=22\\x &=16

3. \overleftrightarrow{S T} is not necessarily the perpendicular bisector of \overline{XY} because not enough information is given in the diagram. There is no way to know from the diagram if \overleftrightarrow{S T} will extend to make a right angle with \overline{XY} .

Practice

For questions 1-4, find the value of x . m is the perpendicular bisector of AB .

m is the perpendicular bisector of \overline{AB} .

  1. List all the congruent segments.
  2. Is C on m ? Why or why not?
  3. Is D on m ? Why or why not?

For Question 8, determine if \overleftrightarrow{S T} is the perpendicular bisector of \overline{XY} . Explain why or why not.

  1. In what type of triangle will all perpendicular bisectors pass through vertices of the triangle?
  2. Fill in the blanks of the proof of the Perpendicular Bisector Theorem.

Given : \overleftrightarrow{C D} is the perpendicular bisector of \overline{AB}

Prove : \overline{AC} \cong \overline{CB}

Statement Reason
1. 1.
2. D is the midpoint of \overline{AB} 2.
3. 3. Definition of a midpoint
4. \angle CDA and \angle CDB are right angles 4.
5. \angle CDA \cong \angle CDB 5.
6. 6. Reflexive PoC
7. \triangle CDA \cong \triangle CDB 7.
8. \overline{AC} \cong \overline{CB} 8.

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