1-7: Midpoint and Distance in the Coordinate Plane

Mean• Average• If you had test scores of 90 and 83, how would you

find your average? 90 83Average

2

86.5

Midpoint• To find the midpoint of a segment on a number line.

Midpoint of AB90 83

2

A B

9083

173

2 86.5

Number Line Midpoint Formula• On a number line, the coordinate of the

midpoint of the segment with endpoints a and b is: 2

a b

Midpoint in a Coordinate Plane

                   

                   

                   

                   

                   

                   

                   

                   

                   

                   

A(1,3)

B(6,8)

(6,3)

6 13.5

2

8 35.5

2

(3.5,5.5)

Coordinate Plane Midpoint Formula• In the coordinate plane, the midpoint of the segment

with endpoints (x1, y1) and (x2, y2) is: 1 2 1 2M ,2 2

x x y y

Ex. 1: Find the coordinates of the midpoint of with the endpoints A (2, 3) and B (-5, 7).

Ex. 2: The midpoint of has coordinates (3, 4). Point A has coordinates (-3, -2). Find the coordinates of B.

2 ( 5),

2M

2 ( 5)

2

AB

AB

,51.5 3 7

2

Distance

A B

To find the horizontal or vertical distance: 6 1 5AB

Apply same idea to finding distance on the Coordinate Plane:

                   

                   

                   

                   

                   

                   

                   

                   

                   

                   

A

B

1x 2x

1y

2y

Distance formula based on Pythagorean Theorem:

2 2 2a b c

a

bc

2 2c a b 2 2

2 1 2 1( ) ( )c x x y y

2 1a x x 2 1b y y

Distance Formula

The distance between two points and is:

1 1,A x y 2 2,B x y

2 22 12 1( ) ( )d yyxx

Ex. 3: Find the distance between each pair of points.a). (1, 2), (3, 4) b). (-1, -2), (-3, -4)

2 23 1 4 2d 2 2

3 ( 1) 4 ( 2)d

4 4d

8 2.83d 8 2.83d

4 4d

Homework: p. 54 #7-39 odd