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

Explanation:

Step1: Recall the properties of dilation

In a dilation, the ratio of the distance from the center of dilation \(Q\) to a point on the image and the distance from the center of dilation \(Q\) to the corresponding point on the pre - image is constant (the scale factor). Also, the ratio of the lengths of corresponding sides of the pre - image and the image is equal to the scale factor.
Since \(QA = 4\) and \(AD=4\), the scale factor \(k=\frac{QD}{QA}=\frac{QA + AD}{QA}=\frac{4 + 4}{4}=2\)

Step2: Check each option

• Option \(QB=\frac{1}{2}QE\):

Let the scale factor be \(k\). If the transformation is a dilation, then \(QE=k\cdot QB\). Since \(k = 2\), \(QB=\frac{1}{2}QE\) (because \(QE = 2QB\))

• Option \(QC=CF\):

Since \(k = 2\), \(QF=k\cdot QC\). And \(QF=QC + CF\). If \(k = 2\), then \(QC+CF=2QC\), which implies \(CF = QC\)

• Option \(DE = 2AB\):

Since \(k = 2\), and \(DE\) and \(AB\) are corresponding sides. The length of the side of the image (\(DE\)) is \(k\) times the length of the corresponding side of the pre - image (\(AB\)). So \(DE=2AB\)

• Option \(AC=\frac{1}{2}EF\):

Since \(k = 2\), \(EF = k\cdot AC\). So \(AC=\frac{1}{2}EF\)

• Option if \(BC = 2.25\), then \(EF = 2.25\):

Since \(k = 2\), if \(BC\) is a side of the pre - image, then \(EF=k\cdot BC\). If \(BC = 2.25\), \(EF=2\times2.25 = 4.5
eq2.25\)

Answer:

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