Question

using the side lengths of △pqr and △stu, which angle has a sine ratio of $\frac{4}{5}$? ∠p ∠q ∠t ∠u

Explanation:

Step1: Recall sine - ratio definition

The sine of an angle in a right - triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse, i.e., $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.
In right - triangle $\triangle PQR$, the hypotenuse $PQ = 20$, and the sides are $PR = 12$ and $QR=16$.
In right - triangle $\triangle STU$, the hypotenuse $TU = 34$, and the sides are $ST = 30$ and $SU = 16$.

Step2: Check ratios for angles in $\triangle PQR$

For $\angle P$ in $\triangle PQR$, $\sin P=\frac{QR}{PQ}=\frac{16}{20}=\frac{4}{5}$.
For $\angle Q$ in $\triangle PQR$, $\sin Q=\frac{PR}{PQ}=\frac{12}{20}=\frac{3}{5}$.

Step3: Check ratios for angles in $\triangle STU$

For $\angle T$ in $\triangle STU$, $\sin T=\frac{SU}{TU}=\frac{16}{34}=\frac{8}{17}$.
For $\angle U$ in $\triangle STU$, $\sin U=\frac{ST}{TU}=\frac{30}{34}=\frac{15}{17}$.

Answer:

$\angle P$