Question

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

Step1: Recall similarity ratio property

For similar triangles, the ratio of perimeters is equal to the scale factor (\(k\)), and the ratio of areas is the square of the scale factor (\(k^2\)). Let the perimeter of \(\triangle DEF\) be \(P_{DEF}\) and of \(\triangle ABC\) be \(P_{ABC}\). Given \(P_{ABC}=5\times P_{DEF}\), so the scale factor \(k = \frac{P_{ABC}}{P_{DEF}} = 5\) (from \(\triangle DEF\) to \(\triangle ABC\)).

Step2: Relate area ratio to scale factor

Let \(A_{ABC}\) be the area of \(\triangle ABC\) and \(A_{DEF}\) be the area of \(\triangle DEF\). The ratio of areas is \(k^2=\frac{A_{ABC}}{A_{DEF}}\). We know \(A_{ABC} = 100\) and \(k = 5\), so \(5^2=\frac{100}{A_{DEF}}\).

Step3: Solve for \(A_{DEF}\)

Simplify \(25=\frac{100}{A_{DEF}}\). Cross - multiply: \(25\times A_{DEF}=100\). Then \(A_{DEF}=\frac{100}{25}=4\).

Answer:

4