Question

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

Explanation:

Step1: Define sin C in △BCD

In right triangle $BCD$ (right-angled at $B$), $\sin C = \frac{\text{opposite side to } C}{\text{hypotenuse}} = \frac{BD}{CD}$

Step2: Use parallel lines for similar triangles

Since $\overline{BC} \parallel \overline{FE}$, $\triangle DFE \sim \triangle DBC$ by AA similarity (shared $\angle D$, corresponding angles from parallel lines are equal). Corresponding angles mean $\angle DFE = \angle B = 90^\circ$, and $\angle DEF = \angle C$.

Step3: Define sin ∠DEF in △DFE

In right triangle $DFE$, $\sin \angle DEF = \sin C = \frac{\text{opposite side to } \angle DEF}{\text{hypotenuse}} = \frac{FD}{ED}$

Answer:

$\boldsymbol{\frac{FD}{ED}}$ (Option: $\boldsymbol{\frac{FD}{ED}}$)