Question

Amias is trying to prove that any two given circles are similar. He starts by drawing circle A with a radius of a and circle B with a radius of b where a≠b. He then translates circle A so that circle A and circle B have the same center. How can he complete his proof to show that circle A is similar to circle B?(1 point)
Responses

Amias can rotate circle A around its center point until it coincides with circle B. Because a sequence of transformations maps circle A onto circle B, circle A is similar to circle B.
Amias can rotate circle upper A around its center point until it coincides with circle upper B . Because a sequence of transformations maps circle upper A onto circle upper B , circle upper A is similar to circle upper B .

Amias can rotate both circle A and circle B around their center points until they coincide. Because a sequence of transformations maps circle A onto circle B, circle A is similar to circle B.
Amias can rotate both circle upper A and circle upper B around their center points until they coincide. Because a sequence of transformations maps circle upper A onto circle upper B , circle upper A is similar to circle upper B .

Amias can dilate circle A by a factor of ab and show that the circles now coincide. Because a sequence of transformations maps circle A onto circle B, circle A is similar to circle B.
Amias can dilate circle upper A by a factor of Start Fraction a over b End Fraction and show that the circles now coincide. Because a sequence of transformations maps circle upper A onto circle upper B , circle upper A is similar to circle upper B .

Amias can dilate circle A by a factor of ba and show that the circles now coincide. Because a sequence of transformations maps circle A onto circle B, circle A is similar to circle B.

Answer 1

To prove that any two given circles are similar, Amias can use the process of dilation, which would effectively change the radius of one circle to match the radius of the other. Here’s how he can complete his proof:

Amias can dilate circle A by a factor of \( \frac{b}{a} \) (the ratio of the radii of the circles) and show that the circles now coincide. Because a sequence of transformations maps circle A onto circle B, circle A is similar to circle B.

Thus, the correct response from the choices given would be:

- "Amias can dilate circle A by a factor of \( \frac{b}{a} \) and show that the circles now coincide. Because a sequence of transformations maps circle A onto circle B, circle A is similar to circle B."

This demonstrates that circles of different radii are similar to each other because similarity in geometry includes the notion of dilation (scaling) without altering the fundamental shapes.