The image shows a problem involving the dilation of triangle $\Delta ABC$ to triangle $\Delta A'B'C'$ using a dilation factor and a center of dilation.

Given Information:

• Dilation Center: $(2, 3)$
• Dilation Factor: $\frac{1}{4}$

The goal is to find the coordinates of point $A'$, the image of point $A$, after dilation.

Steps to Solve:

1. Identify Coordinates of Point A: Based on the grid, the coordinates of point $A$ are approximately $(-2, 6)$.

2. Apply Dilation Formula: The formula for dilation is: $A'(x', y') = (x_c + k(x - x_c), y_c + k(y - y_c))$ where:

   • $(x_c, y_c)$ is the center of dilation
   • $(x, y)$ is the original point (in this case, point $A$)
   • $k$ is the dilation factor

   For point $A$, with coordinates $(-2, 6)$, center $(2, 3)$, and factor $k = \frac{1}{4}$: $x' = 2 + \frac{1}{4}(-2 - 2) = 2 + \frac{1}{4}(-4) = 2 - 1 = 1$ $y' = 3 + \frac{1}{4}(6 - 3) = 3 + \frac{1}{4}(3) = 3 + 0.75 = 3.75$

So, the coordinates of $A'$ are $(1, 3.75)$.

Let me know if you'd like further details or explanations! Here are five related questions to explore:

1. How would the dilation formula change if the center of dilation was not given?
2. How would you find the coordinates of points $B'$ and $C'$ after dilation?
3. How does the size of the dilation factor affect the resulting triangle's size?
4. What is the difference between dilation and other transformations like translation or rotation?
5. Can a dilation change the shape of a figure, or only its size?

Tip: Always remember that dilation affects distances between points, scaling them by the dilation factor relative to the center of dilation.