find the tangent of $angle q$.\nsimplify your answer and write it as a proper fraction, improper fraction…

Answer:

$\frac{15}{8}$

Explanation:

Step1: Find length of $RT$

First, use Pythagorean theorem in $\triangle SRT$: $$RT = \sqrt{34^2 - 16^2} = \sqrt{1156 - 256} = \sqrt{900} = 30$$

Step2: Identify height for $\angle Q$

The height corresponding to $\angle Q$ is $ST = 30$, since $ST \perp SQ$.

Step3: Find base length for $\angle Q$

First, note $\triangle SRT \cong \triangle RQU$ (by AAS, as $\angle S = \angle QRU = 90^\circ$, $\angle RTS = \angle QUR$, $RT=30$), so $SQ = SR + RQ = 16 + 16 = 32$.

Step4: Calculate $\tan(Q)$

Use tangent definition: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ $$\tan(Q) = \frac{ST}{SQ} = \frac{30}{32} = \frac{15}{8}$$