Question

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

Explanation:

Step1: Properties of Dilation

Dilation is a transformation that changes the size of a figure but not its shape. If the scale factor \(n\) satisfies \(0 < n<1\), it is a reduction. Here \(n = \frac{2}{5}\), so \(0<\frac{2}{5}<1\), so the image is a reduction.

Step2: Side - length Relationship

The scale factor \(n=\frac{2}{5}\) means that the side lengths of the image are \(n\) times the side lengths of the pre - image. So the side lengths of the image are \(\frac{2}{5}\) (two - fifths) the size of the corresponding side lengths of the pre - image. For a rectangle, the base is a side length. So the base of the image is \(\frac{2}{5}\) the size of the base of the pre - image.

Step3: Angle Relationship

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

Step4: Center of Dilation

The notation \(D_{V,\frac{2}{5}}\) implies that the center of dilation is at point \(V\) (not \(Q\)), so the statement “The center of dilation is at point \(Q\)” is wrong.

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.