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# Midpoint and Distance Formulas

## 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(x1, y1) and B(x2, y2) 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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