Question

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

(image of right triangle srq with right angle at r, and altitude rp from r to hypotenuse sq, meeting at p)

answer
\\(\bigcirc\\) \\(\frac{rq}{sr}\\) \\(\bigcirc\\) \\(\frac{sr}{sq}\\)
\\(\bigcirc\\) \\(\frac{pq}{rq}\\) \\(\bigcirc\\) \\(\frac{sr}{pr}\\)

Explanation:

Step1: Recall cosine definition

In a right triangle, $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$. For $\angle S$ in $\triangle SRQ$ (right-angled at $R$), adjacent side to $\angle S$ is $SR$, hypotenuse is $SQ$. So $\cos S = \frac{SR}{SQ}$.

Step2: Analyze similar triangles

Since $RP$ is an altitude in right $\triangle SRQ$, $\triangle SRP \sim \triangle SQR$ (by AA similarity, as $\angle S$ is common and both have right angles). In similar triangles, corresponding angles are equal, so $\cos S$ in $\triangle SRP$ should also match. But the key is the original triangle's cosine ratio.

Answer:

$\boldsymbol{\frac{SR}{SQ}}$ (matching the option $\frac{SR}{SQ}$)