### Midpoints and Segment Bisectors

When two segments are congruent, we indicate that they are congruent, or of equal length, with segment markings, as shown below:

A **midpoint** is a point on a line segment that divides it into two congruent segments.

Because

When points are plotted in the coordinate plane, we can use a formula to find the midpoint between them.

Here are two points, (-5, 6) and (3, 2).

The midpoint should be halfway between the points on the segment connecting them. Just by looking, it seems like the midpoint is (-1, 4).

**Midpoint Formula:** For two points,

Let’s use the formula to make sure (-1, 4) is the midpoint between (-5, 6) and (3, 2).

A **segment bisector** cuts a line segment into two congruent parts and passes through the midpoint. A **perpendicular bisector** is a segment bisector that intersects the segment at a right angle.

What if you were given the coordinates of two points and you wanted to find the point exactly in the middle of them? How would you find the coordinates of this third point?

### Examples

#### Example 1

Write all equal segment statements.

#### Example 2

Is

No, it is not

#### Example 3

Find the midpoint between (9, -2) and (-5, 14).

Plug the points into the formula.

#### Example 4

Which line is the perpendicular bisector of

The perpendicular bisector must bisect

#### Example 5

Find

The line shown is the perpendicular bisector.

### Review

- Copy the figure below and label it with the following information:

For 2-4, use the following picture to answer the questions.

P is the midpoint of what two segments?- How does
VS¯¯¯¯¯¯¯ relate toQT¯¯¯¯¯¯¯¯ ? - How does
QT¯¯¯¯¯¯¯¯ relate toVS¯¯¯¯¯¯¯ ?

For exercise 5, use algebra to determine the value of

For questions 6-10, find the midpoint between each pair of points.

- (-2, -3) and (8, -7)
- (9, -1) and (-6, -11)
- (-4, 10) and (14, 0)
- (0, -5) and (-9, 9)
- (-3, -5) and (2, 1)

Given the midpoint

A(−1,2) andM(3,6) B(−10,−7) andM(−2,1)

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.4.