Question

using the side lengths of △pqr (right - angled at r, pr = 12, rq = 16, pq = 20) and △stu (right - angled at s, st = 30, su = 16, tu = 34), which angle has a sine ratio of \\(\frac{4}{5}\\)? \\(\circ\\) \\(\angle p\\) \\(\circ\\) \\(\angle q\\) \\(\circ\\) \\(\angle t\\) \\(\circ\\) \\(\angle u\\)

Explanation:

Step1: Recall Sine Ratio Definition

Sine of an angle in a right triangle is $\frac{\text{opposite}}{\text{hypotenuse}}$.

Step2: Analyze $\triangle PQR$ (Right at $R$)

• For $\angle P$: Opposite = $16$, Hypotenuse = $20$. So $\sin P = \frac{16}{20} = \frac{4}{5}$.
• For $\angle Q$: Opposite = $12$, Hypotenuse = $20$. $\sin Q = \frac{12}{20} = \frac{3}{5}$.

Step3: Analyze $\triangle STU$ (Right at $S$)

• For $\angle T$: Opposite = $16$, Hypotenuse = $34$. $\sin T = \frac{16}{34} = \frac{8}{17}$.
• For $\angle U$: Opposite = $30$, Hypotenuse = $34$. $\sin U = \frac{30}{34} = \frac{15}{17}$.

Answer:

$\angle P$