Question

question 7 points 4
show that circle $c_1$ with center $(0, 2)$ of radius 4 is similar to circle $c_2$ with center $(0, -5)$ of radius 1 by transforming $c_1$ to $c_2$. find the translation rule and scale factor.

note: use your own words to answer this question.

Explanation:

Step1: Determine Translation Rule

To translate the center of \( C_1(0, 2) \) to \( C_2(0, -5) \), we calculate the vertical shift. The change in \( y \)-coordinate is \( -5 - 2 = -7 \), and the \( x \)-coordinate remains the same. So the translation rule is \( (x, y) \to (x, y - 7) \).

Step2: Calculate Scale Factor

The scale factor \( k \) for similar figures is the ratio of the radii of the image to the original. The radius of \( C_2 \) is \( 1 \) and the radius of \( C_1 \) is \( 4 \). So \( k=\frac{\text{Radius of } C_2}{\text{Radius of } C_1}=\frac{1}{4} \).

Answer:

Translation rule: \((x, y) \to (x, y - 7)\); Scale factor: \(\frac{1}{4}\)