Question

in the diagram of right triangle bcd below, \\(\overline{cb} \parallel \overline{ef}\\). which of the following ratios is equivalent to \\(\cos d\\)?
answer
\\(\frac{fd}{fe}\\) \\(\frac{fe}{fd}\\)
\\(\frac{fe}{ed}\\) \\(\frac{fd}{ed}\\)

Explanation:

Step1: Recall cosine definition

For an acute angle in a right triangle, $\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}$. For $\angle D$ in $\triangle BCD$, this is $\cos D = \frac{DB}{DC}$.

Step2: Use parallel lines property

Since $\overline{CB} \parallel \overline{EF}$, $\triangle DFE \sim \triangle DBC$ by AA similarity (shared $\angle D$, corresponding angles from parallel lines are equal).

Step3: Map cosine to small triangle

In $\triangle DFE$, the adjacent side to $\angle D$ is $FD$, and the hypotenuse is $ED$. By similarity, $\cos D = \frac{FD}{ED}$.

Answer:

$\boldsymbol{\frac{FD}{ED}}$ (corresponding option: $\boldsymbol{\frac{FD}{ED}}$)