ALGEBRA

12.6 The Distance and Midpoint Formulas

1 1( , )x y 2 2( , )x y

2 22 1 2 1( ) ( )d x x y y

The distance d between points and is:

Find the distance between (–3, 4) and (1, –4).

Why? Let’s try an example to find out!

22 )4(413

64168054

(-3, 4).

. (1, -4)

4

8

Pythagorean Theorem!

4√5

The Distance Formula

Examples

1. A (3,5) B (7,8)

d =

Distance of AB = 5

2. C (-7,2) D (-2,-10)

Find the distance between the two points.Leave answers in simplified radical form.

√ (7 – 3)² + (8 – 5)² √16 + 9= =√25 = 5

d = √(-2 +7)²+(-10 – 2)² √25 + 144=

=√169 = 13

Distance of CD = 13

Example

d1 =

Yes. It is a right triangle.

Decide whether the points (6,4), (-3,1) and (9,-5) are vertices of a right triangle.

√(6 + 3)²+ (4 – 1)² √81 + 9= =√90 = 3√10

d2 = √ (-3 - 9)²+(1 + 5)² √144 + 36= = √180 = 6√5

d3 = √ (6 - 9)² + (4 + 5)² √9 + 81= = √90 = 3√10

Now use the Pythagorean Theorem Converse to check. Does the sum of the squares of the two shorter sides equal the square of the longest side?

(3√10)² + (3√10)² = (6√5)²

90 + 90 = 180

short² short² long²

180 = 180

The Midpoint Formula

The midpoint of the segment that joins points (x1,y1) and (x2,y2) is the point

2yy

,2

xx 2121

•

•

(-4,2)

(6,8)

282

,2

64- •(1,5)

How does it work?

4 8

Find the coordinate of the Midpoint of BC.

12

7

●1

4

C

●A B

12 + 4

2 ,7 + 1

2

(8,4)

●

●

B (12,7) C (4,1)

Midpoint:

Exercises

1. A (3,5) B (7,-5)

midpoint: 3+72 ,

5+(-5)2

(5,0)

2. A (0,4) B (4,3)

midpoint: 0+42 ,

4+32 (2, )

7

2

(55,-45)

Exercise

midpoint: 10 + 100

2 ,

There are 90 feet between consecutive bases on a baseball diamond. Suppose 3rd base is located at (10,0) and first base is located at (100,-90). A ball is hit and lands halfway between first base and third base. Where does the ball land?

Sketch it.

home

1st

2nd

3rd(10,0) (100,-90)

●

0 - 90

2

Homeworkpg. 748 #15-45 odd #54,55