Question

pentagon ( a ^ { prime } b ^ { prime } c ^ { prime } d ^ { prime } e ^ { prime } ) is the image of pentagon ( a b c d e ) under a dilation with a scale factor of ( \frac { 3 } { 2 } ). what is the length of segment ( overline { a e } )?

Explanation:

Step1: Find the length of segment \(\overline{A'E'}\)

From the graph, the length of segment \(\overline{A'E'}\) is \(6\) units.

Step2: Use the dilation formula

The formula for dilation is \(A'E'=\text{scale factor}\times AE\). Given the scale factor \(k = \frac{3}{2}\), and \(A'E'=6\). Then \(AE=\frac{A'E'}{k}\).
Substitute \(A'E' = 6\) and \(k=\frac{3}{2}\) into the formula: \(AE=\frac{6}{\frac{3}{2}}\).
Using the rule \(\frac{a}{\frac{b}{c}}=a\times\frac{c}{b}\), we have \(AE = 6\times\frac{2}{3}\).
Calculate \(6\times\frac{2}{3}=4\).

Answer:

\(4\)