Question

dilation ( d_{v,\frac{2}{5}} ) was performed on a rectangle. how does the image relate to the pre - image? select three options
the image is a reduction because ( 0lt nlt1 )
the side lengths of the image are two - fifths the size of the corresponding side lengths of the pre - image
the angles of the image are two - fifths the size of the angles of the pre - image
the center of dilation is at point ( q )
the base of the image is two - fifths the size of the base of the pre - image

Explanation:

Step1: Analyze dilation property for reduction

For a dilation \(D_{n}\), if \(0 < n<1\), the image is a reduction. Here \(n = \frac{2}{5}\) and \(0<\frac{2}{5}<1\).

Step2: Analyze side - length relationship

In a dilation \(D_{n}\), if the pre - image has side length \(l\), the image side length \(l^{\prime}=n\times l\). Since \(n = \frac{2}{5}\), the side lengths of the image are \(\frac{2}{5}\) of the pre - image side lengths.

Step3: Analyze angle property

In a dilation, angles are congruent. So the statement “The angles of the image are two - fifths the size of the angles of the pre - image” is wrong.

Step4: Analyze center of dilation

The problem does not give any information about the center of dilation being at point \(Q\). There is no basis for this claim from the given dilation notation \(D_{\frac{2}{5}}\) (the notation \(D_{n}\) does not specify the center unless further information is given).

Step5: Analyze base (a side) relationship

Since the base is a side of the rectangle, and using the side - length relationship from dilation (\(l^{\prime}=n\times l\) with \(n=\frac{2}{5}\)), the base of the image is \(\frac{2}{5}\) of the base of the pre - image.

Answer:

• The image is a reduction because \(0 < n < 1\)
• The side lengths of the image are two - fifths the size of the corresponding side lengths of the pre - image
• The base of the image is two - fifths the size of the base of the pre - image