9.1 Distance Formula & Midpoint Formula

Distance

Find the distance between the 2 points

2 2 2a b c

What theorem ?Pythagorean

(3, 4)

(0, 0) 3

4

a

bc

Distance Formula:

2 2 23 4 c 29 16 c 225 c

5 c Why not ± ?Distance

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

Ex 1) Find the distance between (2, –1) and (9, 1)

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

2 2(9 2) (1 ( 1))d

(x1, y1) (x2, y2)

49 4d

53d

T.O.O Ex 2) Find the distance between

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

2 2(3 5) ( 5 5)d

(x1, y1) (x2, y2)

2 2( 2) ( 2 5)d

4 20d

2 6d

(5, 5) and (3, 5)

24d

Ex 3) Find the distance between(a, b) and (0, b)

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

2 2(0 ) ( )d a b b

(x1, y1) (x2, y2)

2d a

d a

Midpoint Formula

What does midpoint mean?Middle

(3, 4)

(0, 0)

Midpoint Formula:

Add them up & divide by 2

1 2 1 2,2 2

x x y y

approx

Midpoint is +Distance is –

Ex 4) Find the midpoint of (6, 4) and (2, 8)

1 2 1 2,2 2

x x y y

Midpoint Formula:

6 2 4 8,

2 2

4,6

Ex 5) If the midpoint is (–1, 1) and one endpoint is (–6, 2), find the other endpoint.

2 26 2( 1, 1) ,

2 2

x y

Midpoint = (–1, 1) (x1, y1) = (–6, 2) Find (x2, y2)

261

2

x 22

12

y

22 6 x

24 x22 2 y

20 y

Set each part = to each other

2 2( , ) (4,0)x y

Ex 6) Find the equation in Standard Form of the perpendicular bisector of AB for the points A(–2, 1) and B(1, –3).

Perp. Bisector: Opp. recip. slope & goes through midpoint.

Find the slope of the line.

Find the opp. recip. slope.

3 1

1 2m

Find the midpoint of the 2 points

4

3

3

42 1 1 3

,2 2

1, 12

Perp. Bis. has slope and goes thru point 3

4

1, 12

Slope: Point:

Use slope formula again: 1

1

y ym

x x

3

4

1, 12

12

3 1

4

y

x

12

3 1

4

y

x

123( ) 4( 1)x y

33 4 4

2x y

6 3 8 8x y

6 8 5x y

(2) (2) (2) (2)

S.F.: No FractionsAx + By = CA is (+)

T.O.O. Ex 7) Find the midpoint and distance of (–a, b) and (2a, 4b)

1 2 1 2,2 2

x x y y

Midpoint Formula:

2 4,

2 2

a a b b

5,2 2

a b

2 22 1 2 1( ) ( )d x x y y Distance Formula:

2 2(2 ( )) (4 )d a a b b 2 2(3 ) (3 )a b

2 29 9a b 2 29( )a b 2 23 a b

Homework

#901 Pg. 404 1 – 23 odd

Pg. 405 1 – 4 all