Question

in the diagram of right triangle qrs below, \\(\overline{qr} \parallel \overline{ut}\\). which of the following ratios is equivalent to \\(\tan r\\)?
answer
\\(\frac{ut}{us}\\)
\\(\frac{us}{ts}\\)
\\(\frac{ut}{ts}\\)
\\(\frac{us}{ut}\\)

Explanation:

Step1: Define tan R for △QRS

In right △QRS, $\tan R = \frac{\text{opposite}}{\text{adjacent}} = \frac{QS}{QR}$

Step2: Use parallel lines for similarity

Since $\overline{QR} \parallel \overline{UT}$, △SUT ~ △SQR by AA similarity. Corresponding sides are proportional: $\frac{QS}{QR} = \frac{US}{UT}$

Step3: Match to tan R

Substitute: $\tan R = \frac{US}{UT}$

Answer:

$\boldsymbol{\frac{US}{UT}}$ (Option: $\frac{US}{UT}$)