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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.

Dilate each vertex of triangle ABC using P as the center of dilation and a scale factor of 2. Label the points A'B'C'.

Solve the dilation problem involving triangle ABC with center P and a scale factor of 2. The process involves applying the dilation formula to find the new coordinates for A', B', and C'. Perfect for students in Grades 8-10 learning about geometric transformations.

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