Question

circle a is the result of reflecting circle a across the line l. select all of the correct statements about the unchanged properties of circle a and circle a. choose all answers that apply. a the radii of circle a and circle a have the same lengths. b circle a and circle a have the same circumference. c circle a and circle a have the same area. d none of the above

Explanation:

Step1: Recall reflection property

Reflection is a rigid - motion. Rigid - motions preserve distances.

Step2: Analyze radius

Since reflection preserves distances, the radius of circle \(A\) and circle \(A'\) is the same because the distance from the center to the circle's edge is unchanged.

Step3: Analyze circumference

The formula for the circumference of a circle is \(C = 2\pi r\). Since \(r\) is the same for both circles, the circumferences are equal (\(C_A=2\pi r_A\) and \(C_{A'}=2\pi r_{A'}\), and \(r_A = r_{A'}\)).

Step4: Analyze area

The formula for the area of a circle is \(A=\pi r^{2}\). Since \(r\) is the same for both circles, the areas are equal (\(A_A=\pi r_A^{2}\) and \(A_{A'}=\pi r_{A'}^{2}\), and \(r_A = r_{A'}\)).

Answer:

A. The radii of circle \(A\) and circle \(A'\) have the same lengths
B. Circle \(A\) and circle \(A'\) have the same circumference
C. Circle \(A\) and circle \(A'\) have the same area