Question

jeremy wants to verify that the transformation shown is a dilation. he finds the lengths of segments qa and ad to be 4 units. to verify that the transformation is a dilation, jeremy should also check which of the following? select three options.

qb = \\( \frac { 1 } { 2 } \\)qe
qc = cf
de = 2ab
ac = \\( \frac { 1 } { 2 } \\)ef
if bc = 2.25, then ef = 2.25

Explanation:

• For a dilation, the ratio of the distance from the center of dilation to a point on the image and the corresponding point on the pre - image should be constant. Also, the ratio of the lengths of corresponding sides of the image and the pre - image should be constant.
• If \(QA = 4\) and \(AD=4\), the scale factor \(k=\frac{QA + AD}{QA}=\frac{4 + 4}{4}=2\) (assuming \(Q\) is the center of dilation).
• For the ratio of distances from the center of dilation: If \(Q\) is the center, for point \(B\) and its image \(E\), we should check if \(QB=\frac{1}{2}QE\) (since the scale factor \(k = 2\), \(QE=k\times QB\)).
• For the ratio of side lengths:
• Corresponding sides of the pre - image (triangle \(ABC\)) and image (triangle \(DEF\)) should have a ratio equal to the scale factor. If \(k = 2\), then \(DE = 2AB\) (because \(DE\) corresponds to \(AB\)) and \(AC=\frac{1}{2}EF\) (because \(EF\) corresponds to \(AC\) and \(EF=k\times AC\)).

Answer:

\(\boxed{QB=\frac{1}{2}QE}\), \(\boxed{DE = 2AB}\), \(\boxed{AC=\frac{1}{2}EF}\)