Given any two positive integers m and n (n > m), it is possible to form three numbers a, b and c where: a = n^2 - m^2, b = 2mn, c = n^2 + m^2 These three numbers a, b and c are then known as a "Pythagorean triple" - Junior Cycle Mathematics - Question 7 - 2015

To determine if the triangle is right-angled, we will use the Pythagorean theorem which states that for a right triangle: $c^2 = a^2 + b^2$

Identify the largest side, c, which is 34 in this case. Thus, we check: $c^2 = 34^2 = 1156$

Now calculate: $a^2 + b^2 = 16^2 + 30^2 = 256 + 900 = 1156$

Since both sides are equal: $34^2 = 16^2 + 30^2$

By the Pythagorean theorem, the triangle is confirmed to be right-angled.

To show that the triangle formed by the lengths n^2 - m^2, 2mn, and n^2 + m^2 is right-angled, we analyze the expression:

1. We start with: $x = n^2 - m^2,\ y = 2mn,\ z = n^2 + m^2$

2. We then check: $x^2 + y^2 = (n^2 - m^2)^2 + (2mn)^2$

   Expanding these: $= (n^4 - 2n^2m^2 + m^4) + (4m^2n^2)$

   This simplifies to: $= n^4 + 2n^2m^2 + m^4 = (n^2 + m^2)^2$

3. Thus: $z^2 = (n^2 + m^2)^2$

By Pythagoras' theorem, since: $x^2 + y^2 = z^2$ The triangle is confirmed to be right-angled.