Question

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

Explanation:

Step1: Recall Sine Ratio Definition

In a right - triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse, i.e., \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\).

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

• For \(\angle P\):
• The side opposite to \(\angle P\) is \(RQ = 16\), and the hypotenuse is \(PQ=20\). So, \(\sin\angle P=\frac{16}{20}=\frac{4}{5}\).
• For \(\angle Q\):
• The side opposite to \(\angle Q\) is \(PR = 12\), and the hypotenuse is \(PQ = 20\). So, \(\sin\angle Q=\frac{12}{20}=\frac{3}{5}\).

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

• For \(\angle T\):
• The side opposite to \(\angle T\) is \(SU = 16\), and the hypotenuse is \(TU = 34\). So, \(\sin\angle T=\frac{16}{34}=\frac{8}{17}\).
• For \(\angle U\):
• The side opposite to \(\angle U\) is \(ST = 30\), and the hypotenuse is \(TU = 34\). So, \(\sin\angle U=\frac{30}{34}=\frac{15}{17}\).

Answer:

\(\angle P\)