# How do you determine if the lengths 3, 2sqrt10, sqrt41 form a right triangle?

Jul 4, 2018

The Pythagorean Theorem finds any side of a right triangle given the two other sides. We can use this theorem to see if these lengths of sides form a right triangle.

The Pythagorean Theorem Formula is ${a}^{2} + {b}^{2} = {c}^{2}$, where $a$ and $b$ are the lengths of the sides of the triangle and $c$ is the hypotenuse, or the longest side.

Therefore, we can set up an equation with the longest side being $c$:
${3}^{2} + {\left(2 \sqrt{10}\right)}^{2} = {\left(\sqrt{41}\right)}^{2}$

Simplify:
$9 + 4 \cdot 10 = 41$

$9 + 40 = 41$

$49 = 41$

No, $49$ does NOT equal $41$. Therefore, these lengths do not form a right triangle.