7a 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

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