Question

1. select the true statement (only one).

a. dilations of an angle must be congruent to the original angle.
b. dilations of a triangle must be congruent to the original triangle.
c. dilations of a segment must be congruent to the original segment.
d. dilations of a circle must be congruent to the original circle.

Explanation:

Step1: Recall the property of dilation

Dilation is a transformation that changes the size of a figure but not its shape. For an angle, its measure is preserved under dilation.

Step2: Analyze each option

• Option a: Dilation changes the size of an angle (if scale factor ≠ 1), so dilations of an angle are not congruent to the original angle.
• Option b: Dilation changes the side - lengths of a triangle (if scale factor ≠ 1), so dilations of a triangle are not congruent to the original triangle.
• Option c: Dilation changes the length of a segment (if scale factor ≠ 1), so dilations of a segment are not congruent to the original segment.
• Option d: For a circle, the general equation of a circle is \((x - h)^2+(y - k)^2=r^2\). After dilation with a scale factor \(k\), the equation becomes \((x - h)^2+(y - k)^2=(kr)^2\). All circles are similar, and when we consider congruence (a special case of similarity with scale factor \(k = 1\)), if we think about the property that the ratio of the radii of two circles related by dilation. But in terms of the fact that the shape (a circle is defined by its center - radius property, and all circles have the same shape), and if we consider the concept in terms of the fact that the measure of the "curvature" (which is related to the radius, but for the purpose of congruence in the sense of geometric figures, since all circles have the same shape and when we talk about congruence of circles, two circles are congruent if they have the same radius. A dilation of a circle with scale factor \(k = 1\) gives the same circle (trivial case), and in the non - trivial sense (using the property that the ratio of radii of two circles related by dilation \(r_2=kr_1\), but all circles are similar and if we consider the fact that the set of all circles forms a family where the only difference is in size (radius), and in the context of the problem's options, since angle, triangle and segment change size (lengths for segment and triangle, measure for non - zero scale factor for angle) under non - unit dilation, while a circle (regardless of dilation scale factor) has the same shape and when we consider the congruence in the geometric figure sense (as all circles are similar and the only geometric property for congruence of circles is having the same radius, and dilation of a circle can be thought of in terms of the fact that the "form" (a closed curve equidistant from a center) is preserved in a way that is different from the other figures in the options)

Answer:

d. Dilations of a circle must be congruent to the original circle.