Question

what sequence of transformations will transform a circle with center ( o_1 ) and radius ( \frac{4}{5} ) onto a circle with center ( o_2 ) and radius 8?

a. a translation taking ( o_1 ) to ( o_2 ) followed by a dilation with center ( o_2 ) and scale factor 10.

b. a translation taking ( o_1 ) to ( o_2 ) followed by a dilation with center ( o_2 ) and scale factor ( \frac{4}{5} )

c. a translation taking ( o_1 ) to ( o_2 ) followed by a dilation with center ( o_2 ) and scale factor 8

d. a translation taking ( o_1 ) to ( o_2 ) followed by a dilation with center ( o_2 ) and scale factor ( \frac{1}{10} )

Explanation:

Step1: Analyze Translation

To move the center \( O_1 \) to \( O_2 \), a translation is needed. This aligns the centers of the two circles.

Step2: Analyze Dilation Scale Factor

The original radius is \( \frac{4}{5} \), and the target radius is \( 8 \). The scale factor \( k \) for dilation is found by \( \text{target radius} = k \times \text{original radius} \). So \( 8 = k \times \frac{4}{5} \), solving for \( k \) gives \( k = \frac{8}{\frac{4}{5}} = 8 \times \frac{5}{4} = 10 \). So after translation, a dilation with scale factor 10 (center \( O_2 \)) is needed.

Answer:

A. A translation taking \( O_1 \) to \( O_2 \) followed by a dilation with center \( O_2 \) and scale factor 10.