Question

Prove that (12,8),(-2,6)and(6,0)are the vertices of a right angle triangle

Answer 1

Taking A(12, 8), B(-2, 6) and C(6, 0)

AB² = (12--2)² + (8-6)²
= 196+4
=200

AC² = (12-6)² + (8-0)²
= 36+64
=100

BC² = (-2-6)² + (6-0)²
=64+36
=100

By Pythogoras theorem

Since AB² = AC² + BC², the points (12, 8), (-2, 6) and (6, 0) are vertices of a right angled triangle.

AB is the hypotenuse.

Mid-point of AB = [(12+-2)/2 , (8+6)/2]
= (5, 7)
Let the mid-point be M (5, 7)

AM = √(12-5)² + (8-7)²
= √49+1
= √50
= 5√2

MB = √(5- -2)² + (7-6)²
= √49+1
= 5√2

AM = MB = 5√2

This proves that the midpoint of the hypotenuse is equidistant from the angular points.

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