Question

the dilation of triangle abc using the origin as the center of dilation and a scale factor of 2 forms triangle def. describe the perimeter and area. the perimeter of def is the perimeter of abc. the area of def is the area of abc.

Explanation:

Step1: Recall dilation properties

When a figure is dilated with a scale - factor \(k\) about the origin, if the original figure is \(\triangle ABC\) and the dilated figure is \(\triangle DEF\) with scale - factor \(k = 2\).
The ratio of the perimeters of two similar figures is equal to the scale - factor.
Let \(P_{ABC}\) be the perimeter of \(\triangle ABC\) and \(P_{DEF}\) be the perimeter of \(\triangle DEF\). Then \(\frac{P_{DEF}}{P_{ABC}}=k\). Since \(k = 2\), the perimeter of \(\triangle DEF\) is \(2\) times the perimeter of \(\triangle ABC\).

Step2: Recall area - dilation relationship

The ratio of the areas of two similar figures is equal to \(k^{2}\). Let \(A_{ABC}\) be the area of \(\triangle ABC\) and \(A_{DEF}\) be the area of \(\triangle DEF\). Then \(\frac{A_{DEF}}{A_{ABC}}=k^{2}\). Since \(k = 2\), \(k^{2}=4\), so the area of \(\triangle DEF\) is \(4\) times the area of \(\triangle ABC\).

Answer:

The perimeter of \(\triangle DEF\) is \(2\) times the perimeter of \(\triangle ABC\).
The area of \(\triangle DEF\) is \(4\) times the area of \(\triangle ABC\).