Question

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

Explanation:

Step1: Recall Sine Ratio Definition

Sine of an angle in a right triangle is $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$.

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

• Sides: $PR = 12$, $RQ = 16$, $PQ = 20$ (hypotenuse, since $12^2 + 16^2 = 144 + 256 = 400 = 20^2$).
• For $\angle Q$: Opposite side to $\angle Q$ is $PR = 12$, hypotenuse $PQ = 20$. So $\sin(\angle Q)=\frac{12}{20}=\frac{3}{5}$.
• For $\angle P$: Opposite side to $\angle P$ is $RQ = 16$, hypotenuse $PQ = 20$. So $\sin(\angle P)=\frac{16}{20}=\frac{4}{5}$. Wait, no, wait: Wait, $\frac{16}{20}$ simplifies to $\frac{4}{5}$? Wait, $16/20 = 4/5$? Wait, no, $16\div4=4$, $20\div4=5$, yes. Wait, but let's check $\triangle STU$ (Right at $S$).

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

• Sides: $ST = 30$, $SU = 16$, $TU = 34$ (hypotenuse, since $16^2 + 30^2 = 256 + 900 = 1156 = 34^2$).
• For $\angle T$: Opposite side to $\angle T$ is $SU = 16$, hypotenuse $TU = 34$. So $\sin(\angle T)=\frac{16}{34}=\frac{8}{17}$.
• For $\angle U$: Opposite side to $\angle U$ is $ST = 30$, hypotenuse $TU = 34$. So $\sin(\angle U)=\frac{30}{34}=\frac{15}{17}$.

Wait, earlier in $\triangle PQR$, $\angle Q$: opposite is $PR = 12$, hypotenuse $20$, so $\sin Q = 12/20 = 3/5$. $\angle P$: opposite is $RQ = 16$, hypotenuse $20$, so $\sin P = 16/20 = 4/5$? Wait, but the question is about $\frac{4}{5}$? Wait, no, wait the problem says $\frac{4}{5}$? Wait, maybe I miscalculated. Wait, let's recheck $\triangle PQR$:

$PR = 12$, $RQ = 16$, $PQ = 20$. So for $\angle Q$: in right triangle $PQR$, right-angled at $R$, so angle $Q$: the sides adjacent to $Q$ is $RQ = 16$, opposite is $PR = 12$, hypotenuse $PQ = 20$. So $\sin Q = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{20} = \frac{3}{5}$.

For $\angle P$: opposite side is $RQ = 16$, hypotenuse $PQ = 20$, so $\sin P = \frac{16}{20} = \frac{4}{5}$. Wait, but the problem is asking for $\frac{4}{5}$? Wait, maybe I made a mistake. Wait, let's check $\triangle STU$ again. $ST = 30$, $SU = 16$, $TU = 34$. For $\angle T$: opposite is $SU = 16$, hypotenuse $34$, so $\sin T = 16/34 = 8/17$. For $\angle U$: opposite is $ST = 30$, hypotenuse $34$, so $\sin U = 30/34 = 15/17$.

Wait, but in $\triangle PQR$, $\angle Q$: $\sin Q = 12/20 = 3/5$, $\angle P$: $\sin P = 16/20 = 4/5$. Wait, but the options are $\angle T$, $\angle P$, $\angle U$, $\angle Q$. So if the sine ratio is $\frac{4}{5}$, then $\angle P$ has $\sin P = 16/20 = 4/5$. Wait, but wait, maybe I mixed up opposite and adjacent. Wait, in right triangle, sine is opposite over hypotenuse. So for angle $P$ in $\triangle PQR$ (right at $R$), the angle at $P$: the sides: $PR$ is adjacent, $RQ$ is opposite, $PQ$ is hypotenuse. So yes, $\sin P = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{RQ}{PQ} = \frac{16}{20} = \frac{4}{5}$. Wait, but let's check the other triangle. Wait, maybe the problem is $\frac{4}{5}$? Wait, maybe I miscalculated. Wait, 16/20 is 4/5, yes. So $\angle P$ has sine ratio 4/5? Wait, but let's check the options. The options are $\angle T$, $\angle P$, $\angle U$, $\angle Q$. So the answer should be $\angle P$? Wait, no, wait maybe I made a mistake. Wait, let's check $\angle Q$: $\sin Q = 12/20 = 3/5$. $\angle P$: 16/20 = 4/5. $\angle T$: 16/34 = 8/17. $\angle U$: 30/34 = 15/17. So the angle with sine ratio 4/5 is $\angle P$? Wait, but wait, maybe the problem is $\frac{4}{5}$? Wait, maybe the original problem has a typo, but according to the given sides, $\angle P$ in $\triangle PQR$ has $\sin P = 16/20 = 4/5$. So the answer is $\angle P$.

Answer:

$\angle P$