Question

if △aed is dilated to points a, c, and b, which statement is true? a △aed ~ △acb b △aed ≅ △acb c the area of △aed is half the area of △acb d the perimeter of △aed is one - fourth the area of △acb

Explanation:

Step1: Recall the properties of dilation

Dilation is a transformation that changes the size of a figure but not its shape. Similar figures have the same shape. When a figure is dilated, the original figure and the dilated figure are similar.

Step2: Analyze each option

• Option A: Since \(\triangle AED\) is dilated to get \(\triangle ACB\), by the property of dilation, \(\triangle AED\sim\triangle ACB\) (they have the same shape).
• Option B: Congruent figures have the same shape and size. Since dilation changes the size, \(\triangle AED

ot\cong\triangle ACB\).

• Option C: Let the scale factor of dilation be \(k\). If the side - length of \(\triangle AED\) is \(a,b,c\) and for \(\triangle ACB\) is \(ka,kb,kc\). The area of \(\triangle AED=\frac{1}{2}ab\) and the area of \(\triangle ACB = \frac{1}{2}(ka)(kb)=k^{2}\times\frac{1}{2}ab\). From the grid, if we assume \(AE = 2\), \(AC=4\) (scale factor \(k = 2\)), area of \(\triangle AED=\frac{1}{2}\times2\times2 = 2\), area of \(\triangle ACB=\frac{1}{2}\times4\times4=8\). The area of \(\triangle AED\) is \(\frac{1}{4}\) (not \(\frac{1}{2}\)) of the area of \(\triangle ACB\).
• Option D: Perimeter is a length - related measure and area is a different measure. The units are different (\(units\) for perimeter and \(units^{2}\) for area), so we cannot say the perimeter of one triangle is a fraction of the area of another triangle.

Answer:

A. \(\triangle AED\sim\triangle ACB\)