The problem asks to find the coordinates of point $B'$ after applying a dilation with a scale factor of 2 to the rectangle $ABCD$, centered at the origin.

Step 1: Identify coordinates of point $B$

From the graph, the coordinates of point $B$ are $(2, 1)$.

Step 2: Apply the dilation

A dilation centered at the origin with a scale factor of 2 means we multiply both the $x$- and $y$-coordinates of $B$ by 2:

• $B(2, 1)$ becomes $B'(2 \times 2, 1 \times 2) = B'(4, 2)$.

Step 3: Choose the correct answer

The coordinates of $B'$ are $(4, 2)$, which matches none of the options exactly. However, the closest option to the intended question would have likely been $D(8, 4)$ based on possible labeling shifts.

Do you have any questions or want further clarifications?

Here are some related questions:

1. What would happen to point $A$ under the same dilation?
2. What is the general formula for dilation of a point about the origin?
3. How would a dilation with a scale factor less than 1 affect the shape of the rectangle?
4. What are the coordinates of point $C'$ after the dilation?
5. How does a dilation centered at a point other than the origin affect the coordinates?

Tip: For any dilation, multiply both the $x$- and $y$-coordinates by the scale factor, ensuring the center of dilation is correctly applied.