Question

given trapezoid abcd ~ trapezoid gfhd. which proportion is correct?
a. $\frac{ab}{gf} = \frac{bc}{df}$

b. $\frac{ad}{dh} = \frac{ab}{hc}$

c. $\frac{hc}{dh} = \frac{ag}{ad}$

d. $\frac{cd}{fg} = \frac{bc}{gh}$

e. $\frac{bc}{fh} = \frac{ad}{gd}$

Explanation:

Since Trapezoid \(ABCD \sim\) Trapezoid \(GFHD\), corresponding sides of similar trapezoids are proportional. Let's analyze each option:

Step 1: Recall Similar Figures Property

For similar figures, the ratio of corresponding sides is equal. So we need to identify corresponding sides of \(ABCD\) and \(GFHD\).

Step 2: Analyze Option E

• In trapezoid \(ABCD\), sides \(BC\) and \(AD\) are corresponding to sides \(FH\) and \(GD\) in trapezoid \(GFHD\) (since the trapezoids are similar, the non - parallel sides and the parallel sides correspond appropriately).
• So, by the property of similar figures, \(\frac{BC}{FH}=\frac{AD}{GD}\) because \(BC\) corresponds to \(FH\) and \(AD\) corresponds to \(GD\) in the similar trapezoids.

Step 3: Analyze other options (briefly)

• Option A: \(AB\) corresponds to \(GF\), but \(BC\) does not correspond to \(DF\), so \(\frac{AB}{GF}

eq\frac{BC}{DF}\).

• Option B: \(AD\) corresponds to \(GD\) (not \(DH\)) and \(AB\) corresponds to \(GF\) (not \(HC\)), so \(\frac{AD}{DH}

eq\frac{AB}{HC}\).

• Option C: \(HC\) is not a corresponding side with \(DH\) in the way the proportion suggests, and \(AG\) and \(AD\) do not form a correct corresponding ratio, so \(\frac{HC}{DH}

eq\frac{AG}{AD}\).

• Option D: \(CD\) corresponds to \(FD\) (not \(FG\)) and \(BC\) corresponds to \(FH\) (not \(GH\)), so \(\frac{CD}{FG}

eq\frac{BC}{GH}\).

Answer:

E. \(\boldsymbol{\frac{BC}{FH}=\frac{AD}{GD}}\)