Question

If m and n are real numbers and m > n, if m² + n², m² – n² and 2 mn are the sides of the triangle, then prove that the triangle is right-angled. (Use the converse of the Pythagoras theorem). Find out two Pythagorian triplets using convenient values of m and n.

Answer 1

Given: m > n, the sides of the triangle are m² + n², m² – n² and 2 mn.

To prove: The triangle has a right angle.

Proof: (m² + n²)² = (m² – n²)² + (2 mn)²

(m²)² + (n²)² + 2 m²n² = (m²)² + (n²)² – 2 m²n² + 4 m²n²

m⁴ + n⁴ + 2 m²n² = m⁴ + n⁴ + 2 m²n²

The fact that both sides are equal.

So, the triangle supplied is a right-angled triangle according to the converse of the Pythagoras theorem.

	m2 + n2	m2 – n2	2 mn	Pythagorean triplets
m = 3 and n = 1 m = 4 and n = 1	32 + 12 = 10 42 + 12 = 17	32 – 12 = 8 42 – 12 = 15	2 × 3 × 1 = 6 2 × 4 × 1 = 8	(6, 8, 10) (8, 15, 17)

Two Pythagorean triplets are, therefore (6, 8, 10) and (8, 15, 17).