Question

△efg is the image of △abg after a dilation centered at point g with a scale factor of 2. which of the following statements must be true? select all that apply.
overline{ab} is parallel to overline{ef}.
overline{ab} is perpendicular to overline{eg}.
overline{ab} is one - third the length of overline{ef}.
points a and b are the midpoints of overline{eg} and overline{fg}, respectively.

Explanation:

Step1: Properties of Dilation

Dilation is a transformation that enlarges or reduces a figure. If \(\triangle EFG\) is the image of \(\triangle ABG\) after a dilation centered at \(G\) with a scale factor of \(2\), then \(\triangle ABG\sim\triangle EFG\) (by the definition of dilation, which preserves the shape of the figure, i.e., the corresponding angles are equal and the sides are in proportion).
For similar triangles, the corresponding sides are parallel. Since \(AB\) and \(EF\) are corresponding sides of similar triangles \(\triangle ABG\) and \(\triangle EFG\) (because of the dilation transformation), \(\overline{AB}\parallel\overline{EF}\).

Step2: Length relationship

Let the scale factor \(k = 2\). If \(\triangle ABG\sim\triangle EFG\) with scale factor \(k=\frac{EF}{AB}=2\) (from the definition of dilation: if \(\triangle ABC\) is dilated to \(\triangle A'B'C'\) with scale factor \(k\), then \(\frac{A'B'}{AB}=k\)), so \(AB=\frac{1}{2}EF\).
Also, since the scale factor \(k = 2\) and the center of dilation is \(G\), by the property of dilation \(GA=\frac{1}{2}GE\) and \(GB=\frac{1}{2}GF\). So, \(A\) is the mid - point of \(EG\) and \(B\) is the mid - point of \(FG\) (by the definition of a mid - point: a point that divides a line segment into two equal parts). There is no information given to suggest that \(\overline{AB}\perp\overline{EG}\).

Answer:

\(\overline{AB}\) is parallel to \(\overline{EF}\); Points \(A\) and \(B\) are the midpoints of \(\overline{EG}\) and \(\overline{FG}\), respectively.