find the cosine of $angle u$.\nsimplify your answer and write it as a proper fraction, improper fraction, or…

Explanation:

Step1: Identify right triangle for ∠U

We use right triangle $\triangle UTS$ (or $\triangle URV$, since they are similar). First, find the adjacent side to $\angle U$ using the Pythagorean theorem on $\triangle RSV$: For right $\triangle RSV$, $RS=5$, $RV=13$, so the adjacent side to $\angle U$ (corresponding side in $\triangle UTS$) is calculated as: $$\sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$$

Step2: Find hypotenuse of $\triangle UTS$

The hypotenuse of $\triangle UTS$ is $US=12$, wait no—correction: For $\triangle UTS$, the side adjacent to $\angle U$ is the leg we found (12), and the hypotenuse of the larger similar triangle? No, directly: In $\triangle URV$, $\cos(U) = \frac{\text{adjacent to } \angle U}{\text{hypotenuse of } \triangle URV}$. Wait, $\triangle UTS \sim \triangle URV$, so $\cos(U)$ is the ratio of adjacent over hypotenuse in either triangle. For $\triangle RSV$, the leg is 12, so in $\triangle URV$, adjacent side to $\angle U$ is 12, hypotenuse is 13? No, wait: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$. In right triangle $\triangle UTS$, adjacent side to $\angle U$ is the vertical leg (length 12), hypotenuse is $US=12$? No, wait, no: $\triangle RSV$: $RS=5$, $RV=13$, so the other leg is 12. Since $\triangle UTS \sim \triangle URV$, the ratio of adjacent to hypotenuse for $\angle U$ is $\frac{12}{13}$.

Step3: Confirm cosine definition

Cosine of an angle in a right triangle is $\frac{\text{adjacent side}}{\text{hypotenuse}}$. For $\angle U$, the adjacent side is the leg of length 12, hypotenuse is 13.

Answer:

$\frac{12}{13}$