22 in the diagram below, $ \triangle c d e $ is the image of $ \triangle c a b $ after a dilation of $ \frac{d e}{a b} $ centered at $ c $.
which statement is always true?
1) $ \sin a=\frac{c e}{c d} $
2) $ \cos a=\frac{c d}{c e} $
3) $ \sin a=\frac{d e}{c d} $
4) $ \cos a=\frac{d e}{c e} $

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Accepted Answer

# Explanation:
## Step1: Recall the definition of sine in a right - triangle
In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. In $\triangle CAB$, $\sin A=\frac{BC}{AC}$. Since $\triangle CDE$ is a dilation of $\triangle CAB$ (so $\triangle CAB\sim\triangle CDE$ by AA similarity, because $\angle B=\angle DEC = 90^{\circ}$ and $\angle C=\angle C$ (common angle)), the ratios of corresponding sides are equal.
## Step2: Analyze the ratios in $\triangle CDE$
In $\triangle CDE$, $\sin\angle CDE=\frac{CE}{CD}$. But $\angle A=\angle CDE$ (corresponding angles of similar triangles $\triangle CAB$ and $\triangle CDE$).

# Answer:
1) $\sin A=\frac{CE}{CD}$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the definition of sine in a right - triangle

Step2: Analyze the ratios in \(\triangle CDE\)

Answer: