4.20: Perpendicular Bisectors
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Intersect line segments at their midpoints and form 90 degree angles with them.
Perpendicular Bisector Theorem
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.

What if you were given \Delta 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\)?
Example \PageIndex{1}
\overleftrightarrow{OQ} is the perpendicular bisector of \overline{MP}.

Which line segments are equal? Find x. Is L on \overleftrightarrow{OQ}? How do you know?
Solution
ML=LP, MO=OP, and MQ=QP.
\begin{align*} 4x+3&=11 \\ 4x&=8 \\ x&=2\end{align*}
Yes, L is on \overleftrightarrow{OQ} because ML=LP (the Perpendicular Bisector Theorem Converse).
Example \PageIndex{2}
Determine if \overleftrightarrow{ST} is the perpendicular bisector of \overline{XY}. Explain why or why not.

Solution
\overleftrightarrow{ST} 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{ST} will extend to make a right angle with \overline{XY}.
Example \PageIndex{3}
If \overleftrightarrow{MO}− is the perpendicular bisector of \overline{LN} and LO=8, what is ON?

Solution
By the Perpendicular Bisector Theorem, LO=ON. So, ON=8.
Example \PageIndex{4}
Find x and the length of each segment.

Solution
\overleftrightarrow{WX}− is the perpendicular bisector of \overline{XZ} and from the Perpendicular Bisector Theorem WZ=WY.
\begin{align*} 2x+11&=4x−5 \\ 16&=2x \\ 8&=x \end{align*}
WZ=WY=2(8)+11=16+11=27.
Example \PageIndex{5}
Find the value of x. m is the perpendicular bisector of AB.

Solution
By the Perpendicular Bisector Theorem, both segments are equal. Set up and solve an equation.
\begin{align*}3x−8&=2x \\ x&=8 \end{align*}
Review
For questions 1-4, find the value of x. m\) is the perpendicular bisector of AB.
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Figure \PageIndex{9} -
Figure \PageIndex{10} -
Figure \PageIndex{11} -
Figure \PageIndex{12}
m is the perpendicular bisector of \overline{AB}.

- List all the congruent segments.
- Is C on m? Why or why not?
- Is D on m? Why or why not?
For Question 8, determine if \overleftrightarrow{ST} is the perpendicular bisector of \overline{XY}\). Explain why or why not.
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Figure \(\PageIndex{14}\)
- In what type of triangle will all perpendicular bisectors pass through vertices of the triangle?
- Fill in the blanks of the proof of the Perpendicular Bisector Theorem.
Figure \PageIndex{15}
Given: \overleftrightarrow{CD} 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. \Delta CDA\cong \Delta CDB | 7. |
8. \overline{AC}\cong \overline{CB} | 8. |
Review (Answers)
To see the Review answers, open this PDF file and look for section 5.2.
Vocabulary
Term | Definition |
---|---|
circumcenter | The circumcenter is the point of intersection of the perpendicular bisectors of the sides in a triangle. |
perpendicular bisector | A perpendicular bisector of a line segment passes through the midpoint of the line segment and intersects the line segment at 90^{\circ}. |
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. |
Additional Resources
Interactive Element
Video: Perpendicular Bisectors Principles - Basic
Activites: Perpendicular Bisectors Discussion Questions
Study Aid: Bisectors, Medians, Altitudes Study Guide
Practice: Perpendicular Bisectors
Real World: Perpendicular Bisectors