In right triangle PQR, angle P is $30°$, angle Q is $60°$, and the length of side PQ is $8$ units. What is the length of side QR?

• A. $4$ units
• B. $8\times \frac{1}{\sqrt{3}}$ units
• C. $8$ units
• D. $8\times \sqrt{3}$ units

Answer: (D)

Verified step by step guidance

1. Identify the given information: In right triangle PQR, angle P = 30°, angle Q = 60°, and side PQ = 8 units. Since the triangle is right-angled, the remaining angle R must be 90°.
2. Determine the position of the sides relative to the angles: Side PQ is opposite angle R (90°), so PQ is the hypotenuse of the right triangle.
3. Recall the properties of a 30°-60°-90° triangle: The sides are in the ratio 1 : \sqrt{3} : 2, where the side opposite 30° is the shortest, opposite 60° is \sqrt{3} times the shortest, and the hypotenuse is twice the shortest side.
4. Use the side ratio to find the length of side QR, which is opposite angle P (30°). Since PQ is the hypotenuse and equals 8 units, the shortest side (opposite 30°) is half of the hypotenuse: \( \frac{8}{2} = 4 \) units.
5. Calculate side QR, which is opposite angle Q (60°), by multiplying the shortest side by \sqrt{3}: \( 4 \times \sqrt{3} \). This gives the length of side QR.