To solve the problem of dilating triangle $\triangle ABC$ with center $P$ using a scale factor of $\frac{1}{2}$, we will follow these steps:

Steps:

1. Understand the dilation:
   Dilation with a center $P$ and scale factor $\frac{1}{2}$ means each vertex of the triangle $A$, $B$, and $C$ will move closer to $P$ by halving the distance from $P$ to each vertex.

2. Find the new coordinates:

   • Measure or determine the coordinates of points $A$, $B$, $C$, and $P$.
   • Use the formula for dilation: $(x', y') = \left( x_p + k(x - x_p),\ y_p + k(y - y_p) \right)$ where:
     • $(x, y)$: Original point
     • $(x_p, y_p)$: Center of dilation
     • $k$: Scale factor ($\frac{1}{2}$)

3. Apply the formula: Substitute the given scale factor $k = \frac{1}{2}$ into the formula for all three points $A, B,$ and $C$ to find their new coordinates $A', B',$ and $C'$.

4. Plot the new triangle:
   Draw the triangle $\triangle A'B'C'$ with the calculated points.

If you provide the coordinates of $A, B, C,$ and $P$, I can calculate the exact positions of $A', B',$ and $C'$ for you. Alternatively, I can describe how to measure from the diagram to proceed. Let me know! 😊

Follow-Up Questions:

1. What are the coordinates of points $A, B, C,$ and $P$?
2. Do you want me to help you measure the distances manually from the diagram?
3. Would you like a detailed example of how to use the dilation formula?
4. Are you familiar with how to draw dilations accurately on paper?
5. Should I explain how to find the midpoint geometrically for $\frac{1}{2}$ dilations?

Tip:

When working with dilation, always check your scale factor to confirm if the new shape gets larger ($k > 1$) or smaller ($0 < k < 1$).