Given right triangle QRS with right angle at S, if $\mathrm{QR}$ is the hypotenuse and $\mathrm{QS}$ is $8$ units long, and angle Q is $30°$, what is the length of $\mathrm{RS}$?

• A. $4$
• B. $8\surd 3$
• C. $16$
• D. $8$

Answer: (A)

Verified step by step guidance

1. Identify the given elements in the right triangle QRS: the right angle is at S, the hypotenuse is QR, side QS is 8 units, and angle Q is 30°.
2. Recall that in a right triangle, the side opposite the 30° angle is half the length of the hypotenuse, and the side adjacent to the 30° angle relates to the other sides via trigonometric ratios.
3. Use the sine function for angle Q: \(\sin(30^\circ) = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{RS}{QR}\).
4. Use the cosine function for angle Q: \(\cos(30^\circ) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{QS}{QR}\), and since QS = 8, express QR in terms of QS and \(\cos(30^\circ)\).
5. Substitute the value of QR back into the sine equation to solve for RS, the length of the side opposite angle Q.