Distance and Mid point

Quick review: The coordinate plane

• 4 Quadrants

• X axis, horizontal

• Y axis, vertical

• Plotting point (3, -2)

start at origin (0,0)

x, go left (-) or right (+)

y, go up(+) or down(-)

I

III

II

IV(3, -2)

Line Segment vs Line• Iines go on forever,

arrows on each end

• Line segments have end points

• The length of a line segment is the distance between the two endpoints

• Often the end points are assigned letters

(4,3)

(-4,-4)A

B

Length of a Line Segment• What is the length of

this line segment?

Count the spaces from one end point to the other

(4,3)

(4,-4)

A

B

7 units

Length of a Line Segment• What about the length

of this one?

Count the spaces from one end point to the other

(-2,-3)

(4,-4)AB

6 units

Length of a Line Segment• When the line segment

is like this, it is not so easy to find the exact length.

• Just for kicks, estimate how long you think it is

• About 8? 9?

• To find this exact length we use the distance formula

(4,3)

(-4,-4)A

B

Distance Formula

• Used to find the distance between two points

2 2distance ( ) ( )x x y y

Example

• Find the distance between A(-4,-4) and B(4,3)

2 2distance ( ) ( )x x y y

A (-4, -4) B (4, 3)

2 2distance (4 ( 4)) (3 ( 4))

2 2distance (8) (7)

distance 64 49 113 10.63014581

Example

• Find the distance between A(-4,-4) and B(4,3)

2 2distance ( ) ( )x x y y

A (-4, -4) B (4, 3)

2 2distance (4 ( 4)) (3 ( 4))

2 2distance (8) (7)

distance 64 49 113 10.63

YOU TRY!!

• Find the distance between:– A. (2, 7) and (11, 9)

– B. (-5, 8) and (2, - 4)

2 2(9) (2) 85

2 2(7) ( 12) 193

Distance Vs Midpoint• Distance is the

length between two points

• Midpoint is the point that is exactly half way between two points.

9.21 units

(.5,0)

Midpoint Formula

• Used to find the center of a line segment

midpoint ,2 2

x x y y

Example

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

midpoint ,2 2

x x y y

A (4, 8)

B (1, 12)

1 4 12 8midpoint ,

2 2

midpoint 5,10

2 2.5,10or

SAD STACKERA way to remember the midpoint formula

•Stack em

•Add em

•Divide by 2

(4 , 8 )

(1, 12)

(5, 20)

2 2

(2.5, 10)

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

YOU TRY!!

• Find the midpoint between:

• A) (2, 7) and (14, 9)

• B) (-5, 8) and (2, - 4)

midpoint = 8,8

-3midpoint = , 2

2