Question

the diameters of two circles are $d_1 = 10$ and $d_2 = 25$. what is the scale factor ($k$) for a dilation mapping the smaller circle onto the larger circle?
a. 3
b. 2
c. 1.5
d. 2.5

two circles have diameters of 8 and 24, and their centers are 5 units apart. what transformations are needed to map the smaller circle onto the larger circle?
a. dilation and translation
b. rotation and dilation
c. translation and reflection
d. reflection and dilation

circle $c_1$ has a radius of 6 and is centered at $(4, - 3)$. circle $c_2$ has a radius of 12 and shares the same center. what is needed to map circle $c_1$ onto circle $c_2$?
a. reflection
b. rotation
c. dilation
d. translation

Explanation:

Step1: Calculate scale - factor for first question

The scale - factor $k$ of a dilation from a smaller circle to a larger circle is given by the ratio of the larger diameter to the smaller diameter. Given $d_1 = 10$ and $d_2=25$, then $k=\frac{d_2}{d_1}=\frac{25}{10}=2.5$.

Step2: Analyze transformations for second question

The diameters of the two circles are 8 and 24, so a dilation is needed to change the size. Also, since their centers are 5 units apart, a translation is needed to move the center of the smaller circle to the center of the larger circle.

Step3: Analyze transformation for third question

Circle $C_1$ has a radius of 6 and circle $C_2$ has a radius of 12 with the same center. To map $C_1$ onto $C_2$, we only need to increase the size of $C_1$, which is achieved by dilation.

Answer:

1. d. 2.5
2. a. Dilation and translation
3. c. Dilation