Question

in the diagram below, $\triangle bca \sim \triangle dea$. which ratio is equivalent to $\sin d$?
answer
$\frac{ae}{ad}$
$\frac{ad}{ae}$
$\frac{ae}{de}$
$\frac{ad}{de}$

Explanation:

Step1: Define sin D in △DEA

In right △DEA, $\sin D = \frac{\text{opposite side to } D}{\text{hypotenuse}} = \frac{AE}{DE}$

Step2: Use similar triangles property

Since $\triangle BCA \sim \triangle DEA$, corresponding angles are equal: $\angle D = \angle C$.
In right △BCA, $\sin C = \frac{AB}{BC}$, which matches the ratio logic of $\sin D$. From the options, $\sin D = \frac{AE}{DE}$.

Answer:

$\boldsymbol{\frac{AE}{DE}}$ (Option: $\boldsymbol{\frac{AE}{DE}}$)