Point A has coordinates ( x 1 , y 1 ), point B has coordinates...

Ibrahim Rosales

Ibrahim Rosales

Answered

2022-07-20

Point A has coordinates ( x 1 , y 1 ), point B has coordinates ( x 2 , y 2 ) andpoint M has coordinates ( x 1 + x 1 2 , y 1 + y 1 2 )
Prove that A M = 1 2 A B

Answer & Explanation

decoratesuw

decoratesuw

Expert

2022-07-21Added 11 answers


The points A ( x 1 , y 1 ) , B ( x 2 , y 2 ) , M ( x m = x 1 + x 2 2 , y m = y 1 + y 2 2 ) are given.
First note that the points A, M and B are colinear (i.e. lie on the same line):
slope of AM = y m y 1 x m x 1 = y 1 + y 2 2 y 1 x 1 + x 2 2 x 1 = y 2 y 1 x 2 x 1 = slope of AB .
The triangles Δ A M D and Δ A B C are similar, because their corresponding angles are equal. Hence:
A M A B = A D A C = A D 2 A D = 1 2 A M = 1 2 A B .
Taniya Burns

Taniya Burns

Expert

2022-07-22Added 4 answers

A M 2 = ( x 1 + x 2 2 x 1 ) 2 + ( y 1 + y 2 2 y 1 ) 2 = ( x 1 + x 2 2 ) 2 + ( y 1 + y 2 2 ) 2 = 1 4 ( ( x 1 + x 2 ) 2 + ( y 1 + y 2 ) 2 ) = 1 4 A B 2 ,
from which it follows that A M = 1 2 A B.

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