Question

circle a has radius 1.25 cm and circle b has radius 3.75 cm. complete the statements relating these two circles. circle a can be dilated onto circle b with a scale factor of blank. the ratio of circumferences of circle b to circle a is blank. the ratio of diameters of circle b to circle a is blank. the two circles are blank.

Explanation:

Step1: Find the scale factor for dilation

The scale factor of dilation from circle A to circle B is the ratio of their radii. The radius of circle B ($r_B$) is 3.75 cm and the radius of circle A ($r_A$) is 1.25 cm. So the scale factor $k = \frac{r_B}{r_A}$.
$k=\frac{3.75}{1.25}=3$

Step2: Ratio of circumferences

The formula for the circumference of a circle is $C = 2\pi r$. The ratio of the circumferences of circle B to circle A is $\frac{C_B}{C_A}=\frac{2\pi r_B}{2\pi r_A}=\frac{r_B}{r_A}$, which is the same as the scale factor. So $\frac{C_B}{C_A}=3$

Step3: Ratio of diameters

The diameter of a circle $d = 2r$. The ratio of the diameters of circle B to circle A is $\frac{d_B}{d_A}=\frac{2r_B}{2r_A}=\frac{r_B}{r_A}$, which is also the scale factor. So $\frac{d_B}{d_A}=3$

Step4: Determine the relationship between the circles

Since one circle can be obtained from the other by dilation (a similarity transformation), the two circles are similar.

Answer:

Scale factor: $3$; Ratio of circumferences: $3$; Ratio of diameters: $3$; The two circles are similar.