In triangle PQR. `angleQ` is a right angle, QR=24, and PR=26. Triangle YXZ is similar to triangle PQR, where vertices X, Y, and Z correspond to vertices P, Q, and R, respectively, and each side of triangle XYZ is `(1)/(2)` the length of the corresponding side of triangle PQR. What is the value of sin Z?

Options

A. `(5)/(13)`
B. `(5)/(12)`
C. `(12)/(13)`
D. `(12)/(5)`

Answer

To solve the problem step by step, we will analyze triangle PQR and then find the sine of angle Z in triangle YXZ.

### Step 1: Identify the given information
In triangle PQR:
- Angle Q is a right angle (90 degrees).
- QR = 24 (this is one leg of the triangle).
- PR = 26 (this is the hypotenuse).

### Step 2: Find the length of side PQ using the Pythagorean theorem
Since triangle PQR is a right triangle, we can use the Pythagorean theorem:
\[ PR^2 = PQ^2 + QR^2 \]
Substituting the known values:
\[ 26^2 = PQ^2 + 24^2 \]
Calculating the squares:
\[ 676 = PQ^2 + 576 \]
Now, subtract 576 from both sides:
\[ PQ^2 = 676 - 576 = 100 \]
Taking the square root gives:
\[ PQ = \sqrt{100} = 10 \]

### Step 3: Determine the sides of triangle YXZ
Since triangle YXZ is similar to triangle PQR and each side of triangle YXZ is half the length of the corresponding side of triangle PQR:
- XY (corresponding to PQ) = \( \frac{10}{2} = 5 \)
- YZ (corresponding to QR) = \( \frac{24}{2} = 12 \)
- XZ (corresponding to PR) = \( \frac{26}{2} = 13 \)

### Step 4: Find the sine of angle Z
In triangle YXZ, we need to find \( \sin Z \). The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse:
\[ \sin Z = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{XY}{XZ} \]
Substituting the values we found:
\[ \sin Z = \frac{5}{13} \]

### Final Answer
Thus, the value of \( \sin Z \) is:
\[ \sin Z = \frac{5}{13} \]

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