Question

How do you use the Pythagorean Theorem to determine if the three numbers could be the measures of the sides of a right triangle: 6, 12, 18?

Answer 1

Check the similar triangle with sides `1`, `2`, `3` against Pythagoras formula to find that this is not a right angled triangle.

Explanation:

Three positive numbers can be the measures of the sides of a right triangle if and only if taken in ascending order they satisfy:

`a^2+b^2=c^2`

Also a triangle is a right triangle if and only if any similar triangle is a right triangle. So you can multiply or divide `a`, `b` and `c` by any non-zero number before applying the test.

In our example, all of the sides are divisible by `6` so let us assign:

`a = 6/6 = 1`

`b = 12/6 = 2`

`c = 18/6 = 3`

We find `a^2+b^2 = 1^2+2^2 = 1+4 = 5 != 9 = 3^2 = c^2`

In fact these side lengths only form a degenerate 'triangle' of zero area, with interior angles `0`, `0` and `pi`