The problem asks to dilate each vertex of triangle $ABC$ using point $P$ as the center of dilation with a scale factor of 2. This means each vertex will be moved away from point $P$ by twice the current distance between the point and the vertices of the triangle.

Let’s follow these steps to find the new coordinates:

1. Identify coordinates:
   • $A$, $B$, and $C$ are the vertices of the triangle, and we need to get their coordinates relative to the grid.
   • P is the center of dilation.

Let’s assume the following approximate coordinates from the grid:

• $P = (4, 3)$
• $A = (3, 6)$
• $B = (2, 2)$
• $C = (6, 2)$

2. Apply the dilation formula: The dilation formula for any point $(x, y)$ with a center of dilation $(x_P, y_P)$ and a scale factor $k$ is: $(x', y') = (x_P + k(x - x_P), y_P + k(y - y_P))$ Here, $k = 2$ and $(x_P, y_P)$ is the coordinate of point $P$.

3. Calculate the new coordinates:

   • For point $A$: $A' = \left( 4 + 2(3 - 4), 3 + 2(6 - 3) \right) = \left( 4 - 2, 3 + 6 \right) = (2, 9)$
   • For point $B$: $B' = \left( 4 + 2(2 - 4), 3 + 2(2 - 3) \right) = \left( 4 - 4, 3 - 2 \right) = (0, 1)$
   • For point $C$: $C' = \left( 4 + 2(6 - 4), 3 + 2(2 - 3) \right) = \left( 4 + 4, 3 - 2 \right) = (8, 1)$

4. Label the points $A'$, $B'$, and $C'$:

   • $A'(2, 9)$
   • $B'(0, 1)$
   • $C'(8, 1)$

The dilated triangle $A'B'C'$ will have its vertices at these new coordinates.

Would you like further details on any of these steps or any clarifications?

Follow-up questions:

1. How do you find the center of dilation for any shape?
2. What happens to the size of a shape when the scale factor is less than 1?
3. How do you calculate the new coordinates of a point after dilation?
4. Can dilation change the orientation of a shape?
5. What is the difference between dilation and translation in transformations?

Tip: When dilating a shape, the center of dilation stays fixed while all other points move either closer or farther depending on the scale factor.