Sure, let me walk you through the solution step-by-step to find the image of the point [tex]\( C(3, -6) \)[/tex] under the dilation with center [tex]\( (0,0) \)[/tex] and scale factor [tex]\( \frac{5}{3} \)[/tex].
1. Identify the given point and the scale factor:
The given point is [tex]\( C(3, -6) \)[/tex], and the scale factor is [tex]\( \frac{5}{3} \)[/tex].
2. Apply the scale factor to the coordinates:
To find the image of the point after the dilation, multiply both the x-coordinate and the y-coordinate of the point [tex]\( C \)[/tex] by the scale factor.
- For the x-coordinate:
[tex]\[
\text{image}_x = 3 \times \frac{5}{3}
\][/tex]
Let's simplify this:
[tex]\[
\text{image}_x = 3 \times \frac{5}{3} = 5
\][/tex]
- For the y-coordinate:
[tex]\[
\text{image}_y = -6 \times \frac{5}{3}
\][/tex]
Let's simplify this:
[tex]\[
\text{image}_y = -6 \times \frac{5}{3} = -10
\][/tex]
3. Combine the coordinates to form the image point:
The image of the point [tex]\( C(3, -6) \)[/tex] under the dilation is [tex]\( (5, -10) \)[/tex].
4. Check the given answer choices:
Compare the result obtained to the given options:
[tex]\[
\begin{aligned}
&\text{A.} \ \left(\frac{18}{5}, \frac{9}{5}\right) \\
&\text{B.} \ (-10, 5) \\
&\text{C.} \ (5, -10) \\
&\text{D.} \ \left(\frac{9}{5}, \frac{18}{5}\right)
\end{aligned}
\][/tex]
The correct option is:
[tex]\[ \text{C.} \ (5, -10) \][/tex]
So, the image of the point [tex]\( C(3, -6) \)[/tex] under the dilation with the scale factor [tex]\( \frac{5}{3} \)[/tex] is [tex]\( (5, -10) \)[/tex].
