1-3 The Distance and Midpoint Formulas

ObjectivesThe student will be able to:

1. Find the distance between two points.

2. Find the midpoint of a segment.

Distance Formula

Used to find the distance between two points on a number line.

or just count the spaces between the two points.

Example:

Find the distance between A and B.

Distance =

= |-4 – 5|

= |-9|

= 9

A B

Distance Formula

Used to find the distance between two points on a coordinate plane.

2 22 1 2 1distance ( ) ( )x x y y

ExampleFind the distance between A(4,8) and B(1,12)

2 22 1 2 1distance ( ) ( )x x y y

A (4, 8)

B (1, 12)

2 2distance (1 4) (12 8) 2 2distance ( 3) (4)

distance 9 16 25 5

YOU TRY!!Find the distance between:

A. (2, 7) and (11, 9)

B. (-5, 8) and (2, - 4)

2 2(9) (2) 85

Midpoint Formula

Used to find the midpoint between two points on a number line.

Midpoint =

Example:Find the midpoint of A and B.

Midpoint =

A B

Midpoint Formula

Used to find the center of a line segment on a coordinate plane.

2 1 2 1midpoint ,2 2

x x y y

ExampleFind the midpoint between A(4,8) and B(1,12)

2 1 2 1midpoint ,2 2

x x y y

A (4, 8)

B (1, 12)

1 4 12 8midpoint ,

2 2

midpoint 5,10

2

YOU TRY!!

Find the midpoint between:

A) (2, 7) and (14, 9)

B) (-5, 8) and (2, - 4)

midpoint = 8,8

-3midpoint = , 2

2

Find the coordinates of an endpoint using the midpoint.

Example:Find the coordinates of J if K(-1, 2) is the midpoint of JL and L has coordinates (3, -5).

Write two equations using the midpoint formula for a line and the coordinates of the midpoint. 1 using the x-coordinates and 1 using the y-coordinates.

= midpoint

Solve for the variable in each equation and your answers will be the coordinates of the endpoint of the segment.

(2) (2)

-3 -3

(2) (2)

+5 +5

What are the coordinates of the endpoint J of the line segment JL?

J(-5, 9)

Example:Find the coordinates of G if P(-5, 10) is the midpoint of EG and E has the coordinates (-8, 6)

(2) (2)

+8 +8

(2) (2)

-6 -6

What are the coordinates of the endpoint G of the line segment EG?

G(-2, 14)

You Try it:Find the coordinates of E if P(-1, 3) is the midpoint of EG and G has the coordinates (5, 6)

(2) (2)

-5 -5

(2) (2)

-6 -6

What are the coordinates of the endpoint E of the line segment EG?

E(-7, 0)

Using Algebra to find measures

Find the measure of PQ if Q is the midpoint of PR.

If Q is the midpoint of PR, then PQ and QR are equal. 1. Set them equal to each other. 2. Solve for the variable. 3. Substitute the answer in to the equation used for PQ and solve.

9y – 2 = 14 + 5y-5y -5y

4y – 2 = 14+2 +2

4y = 16y = 4

PQ = 9y - 2PQ = 9(4) - 2PQ = 36 - 2PQ = 34

What is the measure of PQ? 34

Example:Find the measure of YZ if Y is the midpoint of XZ and XY =2x – 3 and YZ = 27 – 4x.

1. Set them equal to each other.

2x – 3 = 27 – 4x+4x +4x

6x – 3 = 27 +3 +3

6x = 30

x = 5

2. Substitute in to the equation for YZ.

YZ = 27 – 4xYZ = 27 – 4(5)YZ = 27 – 20

YZ = 7

What is the measure of YZ? 7