Question

show that a^4+b^4 ,a^2b^2 and a^4-b^4 are sides of a right angled triangle

Answer 1

Answer:

We must suppose a > b, for otherwise a⁴ - b⁴ is negative and so is not the side of a triangle.

Also, there is a mistake in the question... the second one should be 2a²b².

Now that they're all positive, we just need to check that they satisfy Pythagoras' Theorem.  So the sum of the squares of two of them must be equal to the square of the third.

( a⁴ - b⁴ )² + ( 2a²b² )²

= (a⁴)² + (b⁴)² - 2a⁴b⁴ + 4a⁴b⁴

= (a⁴)² + (b⁴)² + 2a⁴b⁴

= ( a⁴ + b⁴ )²

So by (the converse of ) Pythagoras' Theorem, these three lengths form a right angled triangle.