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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