The third side would be a leg measuring $35$.
Work Step by Step
We can find the third side by using the Pythagorean theorem, which states that $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs of the right triangle and $c$ is the hypotenuse. Let's plug in what we know into the Pythagorean theorem: $12^2 + 37^2 = c^2$ Evaluate the exponents: $144 + 1369 = c^2$ Add to simplify: $1513 = c^2$ If we take the square root of this number, the answer would not be rational, so we cannot be looking for the value of $c$. Let's see if $37$ may be the hypotenuse, so we may be looking for one of the legs of the right triangle. Let's set up the equation such that $37$ becomes the hypotenuse: $12^2 + b^2 = 37^2$ Evaluate the exponents: $144 + b^2 = 1369$ Subtract $144$ from each side of the equation to move constants to one side of the equation: $b^2 = 1225$ Take the positive square root to solve for $b$: $b = 35$ The third side would be a leg measuring $35$.