The side lengths are labeled as follows: a b c The angle opposite the side labeled

Choice D is correct. By the Pythagorean theorem, if a right triangle has a hypotenuse with length $c$ and legs with lengths $a$ and $b$, then $c^2=a^2+b^2$. In the right triangle shown, the hypotenuse has length $c$ and the legs have lengths $a$ and $b$. It's given that $a=4$ and $b=5$. Substituting $4$ for $a$ and $5$ for $b$ in the Pythagorean theorem yields $c^2=4^2+5^2$. Taking the square root of both sides of this equation yields . Since the length of a side of a triangle must be positive, the value of $c$ is $\sqrt{4^2+5^2}$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.