The point halfway between the endpoints of a line segment is called the midpoint. A midpoint divides...

The point halfway between the endpoints of a line segment is called the midpoint.

A midpoint divides a line segment into two equal parts.

In Coordinate Geometry, there is more than one way to determine the midpoint of a line segment.  

Method 1

You may find the midpoint by simply dividing the length of the segment by 2 and counting that value from either of the endpoints.  

Find the midpoints of line segments AB and CD.

The length of line segment AB is 8 (by counting).  The midpoint is 4 units from

either endpoint.  On the graph, this point is (1,4).

The length of line segment CD is 3 (by counting).  The midpoint is 1.5 units

from either endpoint.  On the graph, this point is (2,1.5)

If the line segments are diagonal, more thought must be paid to the solution.  When you are finding the coordinates of the midpoint of a segment, you are actually finding the average (mean) of the x-coordinates and the average (mean) of the y-coordinates.

Method 2

0 10 20 30-10-20-30

a b

2

ba

Midpoint Formula

Midpoint Coordinates =

2,

22121yyxx

The Midpoint Formula works for all line segments:  vertical, horizontal or diagonal.

Example 1 - Find a Midpoint on a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6

Q R

Method 1

The distance from -3 to 4 is 7. Half of 7 is 3.5, which you add to -3. The midpoint is 0.5.

0 1 2 3 4 5 6-1-2-3-4-5-6

Q R

+7

+3.5

Example 1 - Find a Midpoint on a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6

Q R

Method 2

Use the midpoint formula

5.021

243

Example 2 – Find the Coordinates of a Midpoint

Q = 40

P = -20

TEMPERATURE Find the coordinate of the midpoint PQ

The coordinates of P and Q are -20 and 40. Let M be the midpoint of PQ

10or2

202

4020

M

Find the coordinates of M, the midpoint of PQ, for P(-1, 2) and Q(6, 1)

Let P be (x1, y1) and Q be (x2, y2)

2

3,2

5

2

12,

2

61

2,

22121

M

Myyxx

M

Example 3 - A Different Problem

Find the Coordinates of an Endpoint

Find the coordinates of X if Y(-2, 2) is the midpoint of XQ, and Q has the coordinates (2, 8)

Method 2

Let Z be (x2, y2) in the Midpoint Formula

28

,22

)2,2( 11 yxY

Write two equations to find the coordinates of x

1

1

1

6

242

22

x

x

x

1

1

1

4

842

82

y

y

y

A Different Approach to a Different Problem

Find the Coordinates of an Endpoint

Method 1

Find the coordinates of X if Y(-2, 2) is the midpoint of XQ, and Q has the coordinates (2, 8)

First, let’s try to visualize the problem.

This will give you an idea of where X is located.

Y

Q

Example 3 – Use Algebra to Find Measures

3x + 6

2x + 14

A

B

C

A is the midpoint of segment BC.Find x and the measures of ABand AC.

1) (7 , 4), (9, −1) 2) (8, −9), (0, 5)

3) (1, −7), (1, −12) 4) (0, 4), (−4, -12)

5) (−4, 2), (2, −3) 6) (5, 9), (−1, 9)

7) (−7, 8), (−2, −9) 8) (2, −11), (−9, 0)

9) (4, −1), (2, −7) 10) (−4, −6), (3, −6)

11) (14, 0), (−7, 5) 12) (14, −8), (12, −1)

13) (−4, 12), (−7, −2)

ExercisesGiven the endpoints of a line segment, find the midpoint

Given the midpoint and one endpoint of a line segment, find the other endpoint.

1) Endpoint:(−9, −1), midpoint:(8, 14)

2) Endpoint:(10, 12), midpoint:(6, 9)

3) Endpoint:(−8, −10), midpoint:(10, −7)

4) Endpoint:(−11, 9), midpoint:(3, −11)

5) Endpoint:(−2, 7), midpoint:(12, −10)

6) Endpoint:(11, 14), midpoint:(10, 14)

7) Endpoint:(14, −8), midpoint:(5, 8)

8) Endpoint:(−9, 0), midpoint:(10, −7)