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.