If `DeltaPQR `is right angled at `Q, PQ = 12 and /_PRQ = 30^@`, then what is the value of `QR`?

Options

A. `12sqrt(3)`
B. `12sqrt(2)`
C. `12`
D. `24`

Answer

To find the length of \( QR \) in the right-angled triangle \( \Delta PQR \) where \( Q \) is the right angle, \( PQ = 12 \), and \( \angle PRQ = 30^\circ \), we can use the properties of trigonometric ratios.

### Step-by-Step Solution:

1. **Identify the Triangle and Given Values**:
- We have a right triangle \( \Delta PQR \) with \( Q \) as the right angle.
- The length of \( PQ \) (the side opposite to angle \( PRQ \)) is given as \( 12 \).
- The angle \( \angle PRQ \) is \( 30^\circ \).

2. **Use the Tangent Function**:
- In a right triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
- Here, we can use the tangent of angle \( PRQ \):
\[ \tan(\angle PRQ) = \frac{PQ}{QR} \]

3. **Substitute the Known Values**:
- We know \( PQ = 12 \) and \( \angle PRQ = 30^\circ \).
- The tangent of \( 30^\circ \) is \( \frac{1}{\sqrt{3}} \).
- Thus, we can write:
\[ \tan(30^\circ) = \frac{PQ}{QR} \implies \frac{1}{\sqrt{3}} = \frac{12}{QR} \]

4. **Cross-Multiply to Solve for \( QR \)**:
- Rearranging the equation gives:
\[ QR \cdot \frac{1}{\sqrt{3}} = 12 \]
- Cross-multiplying leads to:
\[ QR = 12 \cdot \sqrt{3} \]

5. **Final Answer**:
- Therefore, the length of \( QR \) is:
\[ QR = 12\sqrt{3} \]