The Distance and Midpoint Formula

What is the distance between the 2 points?(-2,6)

(5,3)

How could you find the distance?

The Distance Formula

The distance between any two points with coordinates (x1,y1) an (x2,y2) is given by the following formula:

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

"The Distance Formula" sung to the tune of "On Top of

Old Smokey"

When finding the distanceBetween the two points,Subtract the two x'sThe same for the y's.Now square these two numbers,And find out their sum.When you take the square rootThen you are all done!

Example 1

Find the distance between the points with coordinates (3,5) and (6,4).

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

2 2(6 3) (4 5)d 2 2(3) ( 1)d

9 1d

10d 3.16 units

Example 2

Determine if triangle ABC with vertices

A(-3,4), B(5,2) and C(-1,-5) is an isosceles triangle. (Hint: An isosceles triangle must have at least 2 sides of equal length.)

A(-3,4) B(5,2)

C(-1,-5)

2 2

2 2

(5 3) (2 4)

(8) ( 2)

664+ 8= 4

AB

2 2

2 2

( 1 5) ( 5 2)

( 6) ( 7)

= 36+ 54 89

BC

2 2

2 2

( 1 3) ( 5 4)

(2) ( 9)

= 4+ 58 81

AC

A(-3,4) B(5,2)

C(-1,-5)

BC and AC have the same length so triangle ABC is Isosceles.

What is the midpoint of the line?

(-2,6)

(5,3)

How could you find the midpoint?

The Midpoint Formula

The coordinates of the midpoint of a line segment whose endpoints are (x1, y1) and (x2, y2) are given by the following formula:

1 2 1 2( , ) ,2 2

x x y yx y

"The Midpoint Formula" sung to the tune of "The Itsy

Bitsy Spider"When finding the midpoint of two points on a graph,Add the two x's and cut their sum in half.Add up the y's and divide 'em by a two,Now write 'em as an ordered pairIt’s the middle of the two.

Example 1

Find the midpoint between the points with coordinates (6,2) and (-3,-4).

6 3 2 4( , ) ,

2 2x y

( , ) (1.5, 1)x y

3 2( , ) ,

2 2x y

(1.5, 1)

Example 2 If the coordinate (2, -3) is the

midpoint of line AB and the endpoint A is (5, 4), what is the missing coordinate B?

1 2

2

x xx

52

2

x

4 5 x

1 x

1 2

2

y yy

43

2

y

6 4 y

10 y

(-1, -10)

Example 3

Find the equation of a line that is perpendicular to the midpoint of a line that contains the endpoints (4, 6) and (-2, 4).

Slope of Line:

Slope of Line:

Midpoint:

Equation of Line:

6 4 2

4 2 6

1

3m

3m

4 2 6 4( , ) ,

2(1,5)

2x y

5 3( 1)y x