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