Question

1. two circles have radii 4 and 12, with centers at (1,1) and (-2,5). what transformations are needed to make the smaller circle similar to the larger circle?

a. translation and rotation
b. dilation reflection
c. reflection and translation
d. dilation and translation

Explanation:

Step1: Understand similarity of circles

All circles are similar. To map one circle to another, we can use translation (to move the center) and dilation (to adjust the radius).
Let the radius of the first circle \(r_1 = 4\) and the radius of the second circle \(r_2=12\). The scale factor of dilation \(k=\frac{r_2}{r_1}=\frac{12}{4} = 3\).
We need to translate the center of the smaller circle \((1,1)\) to the center of the larger circle \((- 2,5)\). The translation vector \(\vec{v}=\langle-2 - 1,5 - 1
angle=\langle-3,4
angle\).

Answer:

d. Dilation and translation