Question

Prove that the following points are the vertices of an isosceles right-angled triangle: 

A(–8, –9), В(0, –3) and C(–6, 5)

Answer 1

Given: A(–8, –9), B(0, –3), C(–6, 5).

To Prove: ΔABC is an isosceles right-angled triangle.

Proof [Step-wise]:

1. Compute squared lengths using distance formula.

AB² = (0 – (–8))² + (–3 – (–9))² 

= (8)² + (6)² 

= 64 + 36

= 100

BC² = (–6 – 0)² + (5 – (–3))² 

= (–6)² + (8)² 

= 36 + 64

= 100

AC² = (–6 – (–8))² + (5 – (–9))² 

= (2)² + (14)² 

= 4 + 196

= 200

2. From the values:

AB = BC

= `sqrt(100)`

= 10

So, two sides are equal, ΔABC is isosceles AB = BC.

AB² + BC² = 100 + 100

= 200

= AC²

So, by the converse of Pythagoras the angle between AB and BC (i.e., ∠B) is 90° ΔABC is right-angled at B.

3. Since the triangle is both isosceles with two equal sides and right-angled, it is an isosceles right-angled triangle.

Therefore, A(–8, –9), B(0, –3) and C(–6, 5) are the vertices of an isosceles right-angled triangle.