Question

1. select whether each statement is true or false. a similarity transformation that maps one circle onto another must include a reflection. there is no similarity transformation that will map one circle onto another. all circles are similar.

Explanation:

Step1: Analyze the first statement

A similarity transformation (translation, rotation, dilation) can map one circle onto another without a reflection. For example, translate the center of one circle to the center of another and then dilate (if radii are different). So the first statement is false.

Step2: Analyze the second statement

Since all circles have the same shape (defined by the equation \((x - a)^2+(y - b)^2=r^2\) where \(a,b\) are center coordinates and \(r\) is radius), we can use translation (to move centers) and dilation (to match radii). So there are similarity transformations that map one circle onto another. The second statement is false.

Step3: Analyze the third statement

By the definition of similar shapes (same shape, different size or position), all circles have the same shape (the ratio of circumference to diameter is \(\pi\) for all circles). So all circles are similar. The third statement is true.

Answer:

False for the first statement, False for the second statement, True for the third statement.