Distance and Midpoint Formulas: Derivation and Usage, Schemes and Mind Maps of Pre-Calculus

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Distance Formula

and

Midpoint Formula

Distance Formula

The distance formula is derived

from the Pythagorean theorem

c^2 = a^2 + b^2.

( x 2 , y 2 )

( x 1 , y 1 ) ( x 2 , y 1 )

| y 2 (^)  y 1 |

| x 2 (^)  x 1 |

d

Substituting d for c,

for a,

and for b in the

Pythagorean equation, you get

| x 2 (^)  x 1 |

| y 2 (^)  y 1 |

2 2 1

2 2 1

2 d | x  x | | y  y |

Parentheses can replace

the absolute value symbols

since we are squaring.

2 2 d  ( x 2 (^)  x 1 (^) )  ( y 2 (^)  y 1 )

Taking the principal square

root yields the distance

formula.

2 2 1

2 2 1

2 d ( x  x ) ( y  y )

The Midpoint Formula

If the endpoints of a segment are

and , then the coordinates of the

midpoint are.

( x 1 , y 1 )

( x 2 , y 2 )

x 1 x 2 y 1 y 2

( x 2 , y 2 )

( x 1 , y 1 )

 

  

  

2

, 2

x 1 x 2 y 1 y 2

Midpoint Formula

If the endpoints of a segment are ( x 1 , y 1 ) and ( x 2 , y 2 ), then the

coordinates of the midpoint are

Example: Find the midpoint of a segment whose endpoints

are (5, 6) and (4, 4).

1 2 1 2

^ x^ ^ x^ y^  y 

 ^ ^ ^  

 ^  