Triangle PQR is a right tyriangle with the `90^(@)` angle at vertex Q. The length of side PQ is 25 and the length of side QR is 60. Triangle STU is similar to trianlge PRQ. The vertices S,T,and U correspond to vertices P, Q, and R, respectively. Each side of triangle STU is `1/10` the length of the corresponding side of triangle PRQ. What is the value of `cos angle U` ?

Options

A. `5/13`
B. `5/12`
C. `5/6`
D. `12/13`

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the sides of triangle PQR
Triangle PQR is a right triangle with:
- PQ = 25 (adjacent side to angle R)
- QR = 60 (opposite side to angle R)

### Step 2: Calculate the hypotenuse PR using the Pythagorean theorem
According to the Pythagorean theorem, in a right triangle:
\[ PR^2 = PQ^2 + QR^2 \]
Substituting the values:
\[ PR^2 = 25^2 + 60^2 \]
\[ PR^2 = 625 + 3600 \]
\[ PR^2 = 4225 \]
Taking the square root:
\[ PR = \sqrt{4225} = 65 \]

### Step 3: Determine the cosine of angle U
Since triangle STU is similar to triangle PQR, the corresponding angles are equal. Therefore, we need to find \( \cos U \) which corresponds to \( \cos R \) in triangle PQR.

The cosine of an angle in a right triangle is defined as:
\[ \cos R = \frac{\text{Adjacent side}}{\text{Hypotenuse}} \]
For angle R in triangle PQR:
\[ \cos R = \frac{PQ}{PR} = \frac{25}{65} \]

### Step 4: Simplify the cosine value
We can simplify \( \frac{25}{65} \):
\[ \cos R = \frac{25 \div 5}{65 \div 5} = \frac{5}{13} \]

### Conclusion
Thus, the value of \( \cos U \) is:
\[ \cos U = \frac{5}{13} \]

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