Question

circle ( a ) is the result of rotating circle ( a ) by ( 180^{circ} ) about point ( p ).
select all of the correct statements about the unchanged properties of circle ( a ) and circle ( a ).
circle ( a ) and circle ( a ) have the same circumference.
the radii of circle ( a ) and circle ( a ) have the same lengths.
points ( a ) and ( a ) are both on the ( x )-axis.
none of the above

Explanation:

Step1: Properties of rotation

Rotation is a rigid transformation. Rigid transformations preserve the shape and size of the figure. For a circle, its size is determined by the radius. Since the radius \(r\) is preserved, the circumference \(C = 2\pi r\) is also preserved.

Step2: Analyzing each option

• Option A:

Since \(C=2\pi r\) and \(r\) is unchanged (due to rotation being a rigid transformation), circle \(A\) and circle \(A'\) have the same circumference.

• Option B:

Rotation is a rigid transformation. A rigid transformation (in 2 - D, like rotation) does not change the lengths of line - segments. The radius of a circle is a line - segment from the center to a point on the circle. So, the radii of circle \(A\) and circle \(A'\) have the same lengths.

• Option C:

Let the center of circle \(A\) be \(A(x_1,y_1)\) (here \(x_1=-3,y_1 = 0\)). When we rotate a point \((x,y)\) about a point \(P(x_0,y_0)\) by \(180^{\circ}\), the formula for the new point \((x',y')\) is \(x'=2x_0 - x\) and \(y'=2y_0 - y\). If \(A(-3,0)\) and \(P(1,-2)\), then \(A'\) has coordinates \(x'=2\times1-(-3)=5\) and \(y'=2\times(-2)-0=-4\). So \(A'\) is not on the \(x\) - axis.

Answer:

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