a = m² - n²

a² = (m² - n²)²
= (m²)² - 2m²n² + (n²)²
= m⁴ - 2m²n² + n⁴

b = 2mn

b² = (2mn)²
= 2²m²n²
= 4m²n²

c = m² + n²

c² = (m² + n²)
= (m²)² + 2m²n² + (n²)²
= m⁴ + 2m²n² + n⁴


a² + b²

= m⁴ - 2m²n² + n⁴ + 4m²n²
= m⁴ - 2m²n² + 4m²n² + n⁴
= m⁴ + 2m²n² + n⁴
= c²

Since the three sides of the triangle satisfy the relation of the Pythagorean Theorem, where the sum of squares of the two shorter sides equals the square of the longest side,

the triangle is right-angled.

Note :

①
c is the longer than a because :

Both m and n are positive integers where m > n

This implies m² and n² are positive integers as well and m² > n²


c - a
= m² + n² - (m² - n²)
= 2n² > 0

∴ c > a

The sum of two positive integers is always greater than their difference.

Eg. If m = 5 , n = 4,

5 > 4
5² > 4²
25 > 16


5² + 4² - (5² - 4²)
= 2(4²)
= 2(16)
= 32 > 0


②c is longer than b because :

Since m > n and both are positive integers,
m - n > 0

Then, c - b
= m² + n² - 2mn
= (m - n)²

Since m - n > 0, then (m - n)² also > 0

Therefore, c - b > 0
c > b


So c is the longest side and it is longer than both a and b.

TQ