# Chapter 8 - Right Triangles and Trigonometry - 8-1 The Pythagorean Theorem and It's Converse - Practice and Problem-Solving Exercises - Page 497: 41

The third side would be a leg measuring $84$.

#### 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: $13^2 + 85^2 = c^2$ Evaluate the exponents: $169 + 7225 = c^2$ Add to simplify: $7394 = c^2$ 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 $85$ 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 $85$ becomes the hypotenuse: $13^2 + b^2 = 85^2$ Evaluate the exponents: $169 + b^2 = 7225$ Subtract $169$ from each side of the equation to move constants to one side of the equation: $b^2 = 7056$ Take the positive square root to solve for $b$: $b = 84$ The third side would be a leg measuring $84$.

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