Prove that the triangle with vertices A (1; 1), B (4; 5) and C (5; 4) is isosceles.

Given a triangle with vertices A (1; 1), B (4; 5) and C (5; 4), we need to prove that this is an isosceles triangle. In order to prove, we need to compose the sides of the triangle using the vertices.

AB = ((4 – 1) ^ 2 + (5 – 1) ^ 2) ^ (1/2) = (3 ^ 2 + 4 ^ 2) ^ (1/2) = (9 + 16) ^ (1 / 2) = 25 ^ (1/2) = 5.

AC = ((5 – 1) ^ 2 + (4 – 1) ^ 2) ^ (1/2) = (4 ^ 2 + 3 ^ 2) ^ (1/2) = (16 + 9) ^ (1 / 2) = 25 ^ (1/2) = 5.

BC = ((5 – 4) ^ 2 + (4 – 5) ^ 2) ^ (1/2) = (1 ^ 2 + 1 ^ 2) ^ (1/2) = (1 + 1) ^ (1 / 2) = 2 ^ (1/2).

It is seen that AB = AC. It follows from this that the triangle is isosceles.

Answer: The triangle is isosceles.