A 13-foot ladder leans against a wall with its base 5 feet from the wall. How high up the wall does the ladder reach?

The wall, ground, and ladder form a right triangle with hypotenuse $13$ and one leg $5$. By the Pythagorean theorem the height is $\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$ feet, choice C. This is the familiar $5$-$12$-$13$ triple. Choice D ($18$) adds $13$ and $5$, which is not how the sides of a right triangle relate. Choice A ($8$) subtracts $13-5$ for the same reason. Choice B ($9$) is not produced by any correct method. The leg must be shorter than the $13$-foot hypotenuse, but longer than the $5$-foot base distance here, since the ladder is steep.