Question

1. which of the following statements is true?

figure 3 (triangle pqr with pq as base, angle at p is 60°, angle at r between rq and rs is 30°, rs is perpendicular to pq at s, sq is 10)
options:
sr = 10√3
pr = 10
ps = 10√3
pq = 10

Explanation:

Step1: Analyze right triangle QSR

In $\triangle QSR$, $\angle QSR=90^\circ$, $\angle SRQ=30^\circ$, so $\angle SQR=60^\circ$. Use $\tan$ for $SR$:
$\tan(60^\circ)=\frac{SR}{QS}$
$SR = QS \times \tan(60^\circ) = 10 \times \sqrt{3} = 10\sqrt{3}$

Step2: Verify other options

• For $PR$: In $\triangle PSR$, $\angle PSR=90^\circ$, $\angle SPR=60^\circ$, $SR=10\sqrt{3}$. $\sin(60^\circ)=\frac{SR}{PR} \implies PR=\frac{10\sqrt{3}}{\frac{\sqrt{3}}{2}}=20

eq 10$

• For $PS$: $\tan(60^\circ)=\frac{SR}{PS} \implies PS=\frac{10\sqrt{3}}{\sqrt{3}}=10

eq 10\sqrt{3}$

• For $PQ$: $PQ=PS+QS=10+10=20

eq 10$

Answer:

$SR = 10\sqrt{3}$