Question

in the diagram of right triangle srq below, altitude \\(\overline{rp}\\) is drawn. which of the following ratios is equivalent to \\(\tan s\\)?

answer
\\(\frac{sr}{rq}\\) \\(\frac{rp}{rq}\\) \\(\frac{pr}{sp}\\) \\(\frac{pr}{sr}\\)

Explanation:

Step1: Define tan S in △SRQ

$\tan S = \frac{\text{opposite}}{\text{adjacent}} = \frac{RQ}{SR}$

Step2: Identify similar triangles

△SRP ~ △SQR (AA similarity: ∠S common, right angles)

Step3: Find equivalent ratio

From similarity: $\frac{PR}{RQ} = \frac{SP}{SR} \implies \frac{PR}{SP} = \frac{RQ}{SR} = \tan S$

Answer:

$\frac{PR}{SP}$