To solve this problem, we'll follow the steps of translation and dilation as described:
1. Initial Position:
[tex]\[
B (5, -2)
\][/tex]
2. Translation:
We translate the point 4 units to the right and 2 units down.
- Translating 4 units to the right involves adding 4 to the x-coordinate:
[tex]\[
x_{\text{translated}} = 5 + 4 = 9
\][/tex]
- Translating 2 units down involves subtracting 2 from the y-coordinate:
[tex]\[
y_{\text{translated}} = -2 - 2 = -4
\][/tex]
The new coordinates after translation are:
[tex]\[
(9, -4)
\][/tex]
3. Dilation:
We now dilate this translated point by a factor of 2 using the origin as the center (origin (0,0)).
- To dilate by a factor of 2, we multiply both coordinates of the translated point by 2:
[tex]\[
x_{\text{dilated}} = 9 \times 2 = 18
\][/tex]
[tex]\[
y_{\text{dilated}} = -4 \times 2 = -8
\][/tex]
The resultant coordinates after dilation are:
[tex]\[
(18, -8)
\][/tex]
Thus, the resulting point after the translation and dilation operations is:
[tex]\[
\boxed{(18,-8)}
\][/tex]
From the given options, the correct answer is:
[tex]\[
A. \quad B^{\prime}(18,-8)
\][/tex]
