© William James Calhoun, 2001

6-7: Midpoint of a Line Segment

OBJECTIVE:You will find the coordinates of the midpoint of a line segment in the coordinate plane.

•line segment - a piece of a line defined by two end-points

•midpoint of a line segment - the point half way between the two end points

To find the midpoint, you go half the distance in the x-direction and half the distance in the y-direction.

To find the midpoint is a simple process:(1) Find the average of the x-coordinates,(2) Find the average of the y-coordinates,(3) The midpoint is then (x-average, y-average).

© William James Calhoun, 2001

6-7: Midpoint of a Line Segment

The coordinates of the midpoint of a line segment whose endpoints

are at (x1, y1) and (x2, y2) are given by

6.7.1 MIDPOINT OF A LINE SEGMENT ON A COORDINATE PLANE

1 2 1 2x x y y, .

2 2

B (2, 5)

A (-4, -1)

To get from point A to point B, you would go:6 units right, then

6 units up.

Going half-way would be:3 units right, then

3 units up.

So the midpoint would be the blue dot at (-1, 2).Notice: (-4 + 2)/2 = -1, and (-1 + 5)/2 = 2.

M (-1, 2)

© William James Calhoun, 2001

EXAMPLE 1: Find the coordinates of the midpoint of a segment with the endpoints (3, -8) and (-4, 2).

1 2 1 2x x y y,

2 2

Use the formula:

midpoint =

3 ( 4) 8 2,

2 2

1 6,

2 2

1, 3

2

That is it.

6-7: Midpoint of a Line Segment

© William James Calhoun, 2001

6-7: Midpoint of a Line Segment

EXAMPLE 2: Find the coordinates of the other endpoint of a segment if one endpoint is (2, 6) and the midpoint is (-4, 2).

The problem gives the x1, y1, and the average of the x’s and y’s.From the formula:

1 2x xx average

2

and 1 2y y

y average2

224

x

2

Solve for x2. Solve for y2.

262

y

2

22 x2 2 4

2

22 x 8 -2 -2

2x 10

26 y2 2 2

2

26 y 4 -6 -6

2y 2

Take these values and write them as an ordered pair.

Answer: (-10, -2).

© William James Calhoun, 2001

6-7: Midpoint of a Line Segment

EXAMPLE 2: Find the coordinates of the other endpoint of a segment if one endpoint is (2, 6) and the midpoint is (-4, 2).

Yes, it is the same example. Here is the CPM of finding the other endpoint.

Endpoint #1 (2, 6)

Midpoint (-4, 2)

Endpoint #2 ( , )

Look at the x-values.To get to -4 from 2, what must you add or subtract?

subtract 6-6Do it again.

-6

-10

Look at the y-values.To get to 2 from 6, what must you add or subtract?

subtract 4Do it again.

-4

-4

-2

There is your answer.

© William James Calhoun, 2001

6-7: Midpoint of a Line Segment

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