Question

Is a triangle with sides of lengths 24, 70 and 74 a right triangle?

Answer 1

Yes

Explanation:

`24^2+70^2=576+4900=5476=74^2`

So these side lengths satisfy Pythagoras Theorem.

An alternative way of approaching this might be as follows:

`24`, `70` and `74` are all divisible by `2`, so this triangle is a right angled triangle if and only if a triangle with sides `12`, `35` and `37` is a right angled triangle.

Notice that `35 = 6^2-1`, `37 = 6^2+1` and `12 = 2*6`. This looks like a pattern we could check:

`(a^2-1)^2 = a^4-2a^2+1`

`(a^2+1)^2 = a^4+2a^2+1`

`(2a)^2 = 4a^2`

So:

`(a^2+1)^2 = a^4+2a^2+1 = a^4-2a^2+1+4a^2 = (a^2-1)^2+(2a)^2`

In our case `a=6`, but any number `a` would work.