Question

in the diagram of right triangle stu below, \\(\overline{ts} \parallel \overline{vw}\\). which of the following ratios is equivalent to \\(\cos u\\)?

diagram of right triangle stu with right angle at s, vw parallel to ts, v on tu, w on su

answer
\\(\frac{st}{su}\\) \\(\frac{su}{tu}\\)
\\(\frac{st}{tu}\\) \\(\frac{su}{st}\\)

Explanation:

Step1: Recall cosine definition in right triangles

In a right triangle, $\cos\theta = \frac{\text{adjacent side to } \theta}{\text{hypotenuse}}$. For $\angle U$ in $\triangle STU$ (right - angled at $S$), the adjacent side to $\angle U$ is $SU$ and the hypotenuse is $TU$. So, $\cos U=\frac{SU}{TU}$.

Step2: Analyze the similar triangles (optional, but to confirm)

Since $\overline{TS}\parallel\overline{VW}$, $\triangle STU\sim\triangle VWU$ (by AA similarity, as both are right - angled and share $\angle U$). But the key is the definition of cosine in the original right triangle $\triangle STU$. The cosine of $\angle U$ is the ratio of the adjacent side ($SU$) to the hypotenuse ($TU$).

Answer:

$\frac{SU}{TU}$ (assuming one of the options is $\frac{SU}{TU}$, based on the analysis of the cosine definition in right - triangle trigonometry)