Question

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

A(–1, –6), В(–9, –10) and C(–7, 6)

Answer 1

Given: A(–1, –6), B(–9, –10) and C(–7, 6).

To Prove: Triangle ABC is right-angled.

Proof [Step-wise]:

1. Compute the squared lengths of the sides using distance formula.

AB = B – A

= (–9 – (–1), –10 – (–6))

= (–8, –4)

AB² = (–8)² + (–4)² 

= 64 + 16

= 80

AC = C – A

= (–7 – (–1), 6 – (–6)) 

= (–6, 12)

AC² = (–6)² + 12² 

= 36 + 144 

= 180

BC = C – B

= (–7 – (–9), 6 – (–10)) 

= (2, 16)

BC² = 2² + 16²

= 4 + 256

= 260

2. Check the Pythagorean relation:

AB² + AC² = 80 + 180 

= 260

= BC²

3. By the converse of the Pythagorean theorem, if the square of one side equals the sum of the squares of the other two sides, the triangle is right-angled.

4. Since AB² + AC² = BC², the angle opposite BC, i.e. ∠BAC is 90°. 

Thus, triangle ABC is right-angled at A.

Triangle ABC is a right-angled triangle with the right angle at A (∠BAC = 90°).