7 A triangle is right-angled if the sides are a = m2 - n2, b = 2mn and c = m2 + n2 where m and n are positive integers, and m > n.

Show that this is true by substituting into the equation c2 = a2 + b2.

The general formula for the sides of right triangle is

a² + b² = c²

Subtitute the equation of a and b to the formula a² + b², evaluate if the answer will be the same as the equation of c²

a² + b²

= (m² - n²) ² + (2mn) ²

= (m² - n²) (m² - n²) + (2mn) (2mn)

= (m⁴ - 2m²n² + n⁴) + 4m²n²

= m⁴ + 2m²n² + n⁴

Factorize the result

m⁴ + 2m²n² + n⁴

= (m² + n²) (m² + n²)

= (m² + n²) ²

= c²

It's proven by the formula that c² = (m² + n²) ²

so the c will be m² + n²