Question

1. which is a true statement about dilations?

a dilation always changes the orientation of a figure.
a dilation sometimes changes the orientation of the vertices.
a dilation sometimes enlarges a figure.
a dilation always preserves congruence.

Explanation:

Step1: Recall Dilation Properties

Dilation is a transformation that changes the size of a figure (scaling) about a center point. It does not change the shape (similar figures) or orientation (direction of vertices/figure). Congruence is preserved only if the scale factor is \(1\) (identity dilation), otherwise, the figure is similar (not congruent) with changed size.

Step2: Analyze Each Option

• Option 1: "A dilation always changes the orientation of a figure." → Dilation preserves orientation (e.g., a triangle's vertices' order/direction remains same). So this is false.
• Option 2: "A dilation sometimes changes the orientation of the vertices." → Dilation never changes vertex orientation (it's a similarity transformation preserving orientation). So this is false.
• Option 3: "A dilation sometimes enlarges a figure." → If the scale factor \(k > 1\), dilation enlarges; if \(0 < k < 1\), it reduces; if \(k = 1\), size is same. So "sometimes" (when \(k > 1\)) it enlarges. This is true.
• Option 4: "A dilation always preserves congruence." → Congruence is preserved only when \(k = 1\). For \(k

eq1\), the figure is similar (not congruent) with different size. So this is false.

Answer:

A dilation sometimes enlarges a figure. (The option with the blue circle, which is the third option: "A dilation sometimes enlarges a figure.")