https://file.aiquickdraw.com/mathbot/file/17260572554292majyoyb.jpg

The image shows a right triangle \( \triangle PQR \), where angle \( \angle PQR = 90^\circ \), and the lengths of the sides are to be calculated. The given information includes:

- \( \angle QRP = 60^\circ \),
- \( \angle PRQ = 30^\circ \),
- \( PQ = 18 \) (the hypotenuse).

To find the lengths of sides \( PR \) and \( QR \), we can use trigonometric functions since this is a right triangle.

### 1. To find \( PR \) (opposite \( 30^\circ \)):
\[
\sin(30^\circ) = \frac{PR}{PQ}
\]
\[
PR = PQ \times \sin(30^\circ) = 18 \times 0.5 = 9
\]

### 2. To find \( QR \) (adjacent to \( 30^\circ \)):
\[
\cos(30^\circ) = \frac{QR}{PQ}
\]
\[
QR = PQ \times \cos(30^\circ) = 18 \times \frac{\sqrt{3}}{2} = 18 \times 0.866 = 15.59
\]

### Answer:
- \( PR = 9 \)
- \( QR = 15.59 \)

Would you like more details, or have any other questions?

Here are 5 related questions:
1. How would you solve this problem using the Pythagorean theorem?
2. What are the key trigonometric ratios for 30°, 60°, and 90° triangles?
3. How can you calculate angles given side lengths?
4. What is the significance of special triangles like 30-60-90 triangles in geometry?
5. Can you derive the sine and cosine of 60°?

**Tip**: When working with right triangles, knowing one angle and the hypotenuse often simplifies solving for the other sides.

Learn how to calculate the lengths of sides PQ and QR of a right triangle using trigonometric functions. This problem covers key concepts in trigonometry, including sine and cosine ratios for special angles 30° and 60°, and is suitable for students in Grades 8-10.

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