Question

quadrilateral abcd is similar to quadrilateral abcd.
select all statements that must be true.
a (\frac{ab}{ab}=\frac{ac}{ac})
b (\frac{ad}{ad}=\frac{bc}{bc})
c (\frac{bd}{bd}=\frac{cd}{cd})
d (\frac{ab}{cd}=\frac{ab}{cd})
e (\frac{bc}{ad}=\frac{bc}{ad})

Explanation:

Step1: Recall the property of similar quadrilaterals

For similar quadrilaterals, the ratios of corresponding sides are equal.

Step2: Analyze each option

• Option A:

Since \(A'B'\) corresponds to \(AB\) and \(A'C'\) corresponds to \(AC\), \(\frac{A'B'}{AB}=\frac{A'C'}{AC}\) holds as they are ratios of corresponding sides.

• Option B:

\(AD\) corresponds to \(A'D'\) and \(BC\) corresponds to \(B'C'\). The correct ratio should be \(\frac{AD}{A'D'}=\frac{BC}{B'C'}\), not \(\frac{AD}{A'D'}=\frac{B'C'}{BC}\).

• Option C:

\(BD\) and \(B'D'\) are not corresponding sides in the same proportion as \(CD\) and \(C'D'\). The ratio \(\frac{BD}{B'D'}=\frac{C'D'}{CD}\) is incorrect.

• Option D:

\(AB\) corresponds to \(A'B'\) and \(CD\) corresponds to \(C'D'\). So \(\frac{AB}{CD}=\frac{A'B'}{C'D'}\) is correct as it follows the ratio of corresponding sides.

• Option E:

\(BC\) corresponds to \(B'C'\) and \(A'D'\) corresponds to \(AD\). The ratio \(\frac{BC}{A'D'}=\frac{B'C'}{AD}\) is incorrect.

Answer:

A. \(\frac{A'B'}{AB}=\frac{A'C'}{AC}\), D. \(\frac{AB}{CD}=\frac{A'B'}{C'D'}\)