Question: Rectangle ABCD is dilated using E as the center
of dilation with a scale factor of 3.
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Graph shapes and math:
A, B, C, D form a rectangle with center E inside it; the original rectangle is at center-left of the grid, and the dilated image is positioned by scaling from E by factor 3, with the image rectangle extending farther in the same directions from E.
1. **Problem Statement:**
We have rectangle ABCD with center E inside it. The rectangle is dilated using E as the center of dilation with a scale factor of 3.
2. **Formula for Dilation:**
If $P$ is a point on the original figure and $P'$ is the corresponding point on the dilated image, then
$$P' = E + k(P - E)$$
where $k$ is the scale factor (here $k=3$).
3. **Explanation:**
- The center of dilation $E$ remains fixed.
- Each point $P$ of the rectangle is moved along the line from $E$ to $P$.
- The distance from $E$ to $P'$ is $3$ times the distance from $E$ to $P$.
4. **Intermediate Work:**
Let the coordinates of $E$ be $(x_E,y_E)$ and a point $P$ be $(x_P,y_P)$.
Then the dilated point $P'$ is:
$$x_{P'} = x_E + 3(x_P - x_E)$$
$$y_{P'} = y_E + 3(y_P - y_E)$$
5. **Interpretation:**
- The image rectangle is larger by a factor of 3 in all directions from $E$.
- The shape remains a rectangle because dilation preserves angles and shape.
6. **Final Answer:**
The dilated rectangle $A'B'C'D'$ is obtained by applying the formula above to each vertex $A, B, C, D$ using $E$ as the center and scale factor $3$.