Question

select the correct answer from the drop-down menu.\
$\triangle abc$ is similar to $\triangle def$. the perimeter of $\triangle abc$ is five times the perimeter of $\triangle def$.\
the area of $\triangle abc$ is 100 square centimeters. the area of $\triangle def$ is \\u25bc square centimeters.\
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Explanation:

Step1: Recall similarity ratio properties

For similar figures, the ratio of perimeters is equal to the scale factor (\(k\)), and the ratio of areas is equal to the square of the scale factor (\(k^2\)). Given the perimeter of \(\triangle ABC\) is 5 times that of \(\triangle DEF\), so the scale factor \(k\) (from \(\triangle DEF\) to \(\triangle ABC\)) is 5. Thus, the ratio of areas \( \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle DEF} = k^2 = 5^2 = 25\).

Step2: Solve for area of \(\triangle DEF\)

Let \(A_{DEF}\) be the area of \(\triangle DEF\) and \(A_{ABC} = 100\). We have \(\frac{100}{A_{DEF}} = 25\). Solving for \(A_{DEF}\), we get \(A_{DEF} = \frac{100}{25} = 4\).

Answer:

4