## Key Concepts

- Find segment lengths
- Use algebra with segment lengths
- Use the midpoint formula

### Introduction

In this chapter, we will learn to find segment lengths based on the midpoint, using algebra to find segment lengths, using the midpoint formula and distance formula.

### Midpoints

The midpoint of a segment is the point that divides the segment into two congruent segments.

M is the midpoint of segment AC

M bisects segment AC

### Bisectors

A segment bisector can be a point, ray, line, line segment, or plane that intersects the segment at its midpoint.

A midpoint or a segment bisector bisects a segment.

**Example of segment bisectors: **

### Find segment lengths

**Example 1:**

In the skateboard design, bisects at point T, and = 39.9 cm. Find .

**Solution:**

Point T is the midpoint of XY. So, XT = TY = 39.9cm.

**Example 2:**

Find RS.

**Solution:**

Point T is the midpoint of RS. So RT= TS = 21.7

### Use algebra with segment lengths

**Example 3:**

Point M is the midpoint of VW. Find the length of VW .

**Solution:**

STEP 1: Write and solve an equation. Use the fact that VM = MW

**Example 4:**

Point C is the midpoint of BD. Find the length of BC.

**Solution: **

Step 1: Write and solve an equation.

### Use the Mid Point Formula

#### Midpoint formula

**Example 5:**

- Find midpoint: The endpoints of RS are R(1, 23) and S(4, 2). Find the coordinates of the midpoint M.

**Solution:**

- Find midpoint:

Use the midpoint formula.

The coordinates of the midpoint M are:

- Find endpoint: The midpoint of JK−JK- is M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K.

**Solution:**

- Find endpoint:

Let (x, y) be the coordinates of endpoint K.

Use the midpoint formula.

STEP 1: Find *x*.

1+ x = 4

x = 3

STEP 2: Find *y.*

4 + y = 2

Y = -2

The coordinates of endpoint K are (3, -2).

#### Distance Formula

The distance formula is a formula for computing the distance between two points in a coordinate plane.

If A(x_{1}, y_{1}) and B(x_{2}, y_{2}) are points in a coordinate plane, then the distance between A and B is

**Example 6:**

Find distance between R and S. Round to the nearest tenth if needed.

## Exercise

- What is the difference between these three symbols: = and ≈?
- Identify the segment bisectors of Then find

- Identify the segment bisectors of RS. Then find RS

- Identify the segment bisectors of XY Then find XY

- Identify the segment bisectors of XY Then find XY

- What is the approximate length of AB with endpoints A(-3, 2) and B(1, -4)?
- What is the approximate length of RS with endpoints R(2, 3) and S(4, -1)?
- Find the midpoint of the segment between (20, -14) and (-16, 4).
- The midpoint of segment DH is O(3, 4). One endpoint is D(5, 7). Find the coordinates of H.

- Work with a partner. Use centimeter graph paper.

- Graph AB , where the points A and B are as shown below.
- Explain how to bisect AB , that is, to divide AB into two congruent line segments. Then bisect AB and use the result to find the midpoint M of AB .

- What are the coordinates of the midpoint M?
- Compare the x-coordinates of A, B, and M. Compare the y-coordinates of A, B, and M. How are the coordinates of the midpoint M related to the coordinates of A and B?

### What have we learned

- Finding segment lengths based on midpoint.
- Using algebra to find segment lengths.
- Using the midpoint formula and distance formula.

### Concept Map

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