Question

select all of the correct statements about the unchanged properties of circle a and circle a.
choose all answers that apply:
a points a and a have the same coordinates.
b circle a and circle a have the same area.
c the radii of circle a and circle a have the same lengths.
d none of the above

Explanation:

Step1: Analyze property of coordinate

If a circle is transformed (such as translation, rotation etc.), the center (point \(A\) for circle \(A\) and \(A'\) for circle \(A'\)) may change its coordinate. So, we cannot assume points \(A\) and \(A'\) have the same coordinates.

Step2: Analyze property of area and radius

The area of a circle is given by the formula \(A = \pi r^{2}\), where \(r\) is the radius. If two circles are congruent (same - sized), then \(r = r'\) (where \(r\) is the radius of circle \(A\) and \(r'\) is the radius of circle \(A'\)). Since \(A=\pi r^{2}\) and \(A'=\pi r'^{2}\), when \(r = r'\), \(A = A'\). Congruent circles have radii of the same length.

Answer:

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