# 4.20: Perpendicular Bisectors

- Page ID
- 4817

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\( AB\).

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.

- 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.

__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