# Using the Pythagorean Theorem, is a triangle with side lengths \sqrt 2, \sqrt3, and \sqrt 5 ft a right triangle?

Feb 28, 2018

Yes

#### Explanation:

The theorem states that the length of the sides $a , b \mathmr{and} c$ of a right-angled triangle is related as follows: ${a}^{2} + {b}^{2} = {c}^{2}$ where $c$ is obviously the hypotenuse.

We can use the formula in reverse to check if the given side lengths form a right triangle or not.

If we assume $\sqrt{5}$ to be the hypotenuse of this triangle (by common sense - it is the longest side), we get ${\sqrt{2}}^{2} + {\sqrt{3}}^{2} = {\sqrt{5}}^{2}$

But since ${\sqrt{x}}^{2}$ is $x$ for positive $x$, we can simplify it to: $2 + 3 = 5$.

And that is true indeed.