Question

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

Explanation:

Step1: Recall the sine ratio formula

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\).

Step2: Analyze \(\triangle PQR\)

In \(\triangle PQR\) with \(\angle R = 90^{\circ}\), for \(\angle Q\), the side opposite \(\angle Q\) is \(PR = 12\) and the hypotenuse \(PQ=20\), \(\sin\angle Q=\frac{PR}{PQ}=\frac{12}{20}=\frac{3}{5}\). For \(\angle P\), the side opposite \(\angle P\) is \(QR = 16\) and the hypotenuse \(PQ = 20\), \(\sin\angle P=\frac{QR}{PQ}=\frac{16}{20}=\frac{4}{5}\).

Step3: Analyze \(\triangle STU\)

In \(\triangle STU\) with \(\angle S=90^{\circ}\), for \(\angle T\), the side opposite \(\angle T\) is \(SU = 16\) and the hypotenuse \(TU = 34\), \(\sin\angle T=\frac{SU}{TU}=\frac{16}{34}=\frac{8}{17}\). For \(\angle U\), the side opposite \(\angle U\) is \(ST = 30\) and the hypotenuse \(TU=34\), \(\sin\angle U=\frac{ST}{TU}=\frac{30}{34}=\frac{15}{17}\).

Answer:

\(\angle P\)