Distance Formulaand

Midpoint Formula

Distance Formula

The distance formula is derived from the Pythagorean theorem c2

= a2 + b2.

),( 22 yx

),( 12 yx),( 11 yx

|| 12 yy

|| 12 xx

d

Substituting d for c, for a,

and for b in the Pythagorean equation, you get

|| 12 xx

|| 12 yy

212

212

2 |||| yyxxd

Parentheses can replace the absolute value symbols since we are squaring.

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

Taking the principal square root yields the distance formula.

212

212

2 )()( yyxxd

The distance d between any two points (x1, y1) and (x2, y2) is given by

Example: Find the distance between the points (2, 2) and (3, 5). Click here to check your answer.

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

2 2

2 2

( 3 2) ( 5 2)

( 5) ( 7) 25 49

74 8.6

d

d

d

The Midpoint Formula

If the endpoints of a segment areand , then the coordinates of the midpoint are .

),( 11 yx

),( 22 yx

2,

22121 yyxx

),( 22 yx

),( 11 yx

2,

22121 yyxx

Midpoint FormulaIf the endpoints of a segment are (x1, y1) and (x2, y2), then the coordinates of the midpoint are

Example: Find the midpoint of a segment whose endpoints are (5, 6) and (4, 4). Click here to check your answer.

1 2 1 2, .2 2

x x y y

5 4 6 4,

2 2

1 2,

2 2

1, 1

2