Question

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

A(–2, 2), B(13, 11) and C(10, 14)

Answer 1

Given: A(–2, 2), B(13, 11), C(10, 14)

To Prove: A, B, C are vertices of a right‑angled triangle specify which angle is right.

Proof [Step-wise]:

1. Compute vectors CA and CB vectors from C to A and C to B:

CA = A – C

= –2 – 10, 2 – 14 

= –12, –12

CB = B – C

= 13 – 10, 11 – 14

= 3, –3

2. Compute the dot product CA · CB: 

CA · CB = (–12)(3) + (–12)(–3)

= –36 + 36 

= 0

Since the dot product of CA and CB is 0, vectors CA and CB are perpendicular.

Therefore, lines CA and CB are perpendicular, so ∠ACB = 90°.

3. Compute squared lengths:

CA² = (–12)² + (–12)² 

= 144 + 144

= 288

CB² = 3² + (–3)²

= 9 + 9

= 18

AB² = (13 – (–2))² + (11 – 2)²

= 15² + 9²

= 225 + 81

= 306

4. Verify Pythagoras:

CA² + CB²

= 288 + 18

= 306

= AB² 

Which shows the side AB is the hypotenuse and the right angle is at C since CA² + CB² = AB².

Since CA · CB = 0 and equivalently CA² + CB² = AB², the segments CA and CB are perpendicular and triangle ABC is right‑angled at C.