Question

the pentagons abcde and pqrst are similar. find the length x of $overline{qr}$.

Explanation:

Step1: Identify corresponding sides

For similar pentagons, the ratio of corresponding - side lengths is equal. Consider the ratio of the side lengths of the two pentagons. Let's take the ratio of the sides \(AB\) and \(PQ\). \(AB = 2\) and \(PQ = 1\).

Step2: Set up proportion

The ratio of the side lengths of the two similar pentagons is constant. If we consider the sides \(BC\) and \(QR\), the proportion is \(\frac{AB}{PQ}=\frac{BC}{QR}\). Substituting the known values, we have \(\frac{2}{1}=\frac{4}{x}\).

Step3: Solve the proportion

Cross - multiply the proportion \(\frac{2}{1}=\frac{4}{x}\) to get \(2x = 4\times1\). Then, divide both sides of the equation \(2x = 4\) by 2. So, \(x=\frac{4}{2}=2\).

Answer:

\(x = 2\)