Question

problem 1
which dilation moves triangle gef onto triangle gef?
a dilation of triangle gef using:
center of dilation: point e
scale factor: \\( \frac { 1 } { 2 } \\)
center of dilation: point g
scale factor: \\( \frac { 1 } { 2 } \\)
center of dilation: point g
scale factor: 2
center of dilation: point e
scale factor: 2

Explanation:

Step1: Recall the concept of dilation

Dilation is a transformation that changes the size of a figure. If the center of dilation is a point, and we know the lengths of corresponding sides. Here, \(GF = 8\) units and \(GC=4\) units.

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the ratio of the length of a side of the image to the length of the corresponding side of the pre - image. If we consider the center of dilation as point \(G\), and assume \(\triangle GEF\) is dilated to \(\triangle G'EF'\). The scale factor \(k=\frac{GC}{GF}\) (if we consider the side related to the center). But actually, for a dilation that maps \(\triangle GEF\) to \(\triangle G'EF'\) (where \(G'C = 4\) and \(GF = 8\)), the scale factor \(k=\frac{1}{2}\). And since the center of dilation is a point that remains fixed during the dilation. When we dilate \(\triangle GEF\) with center of dilation at \(G\) and scale factor \(\frac{1}{2}\), the side \(GF\) (length \(8\)) is mapped to a side (say \(G'F\)) such that \(G'C=\frac{1}{2}GF\) (where \(C\) is a point on the dilated side).

Answer:

Center of dilation: point \(G\), Scale factor: \(\frac{1}{2}\)