Let's solve the dilation problem step by step. We'll identify the coordinates of the triangle's vertices, apply the dilation formula, and then determine the coordinates of the new vertices.

Step 1: Identify the Coordinates of the Original Points

From the image, let's identify the coordinates of the vertices of the triangle.

• $A(2, 2)$
• $B(4, 4)$
• $C(6, 1)$

Step 2: Use the Dilation Formula

The center of dilation is $(3, -2)$ and the scale factor $k$ is $\frac{1}{2}$.

The dilation formula for a point $(x, y)$ is: $(x', y') = \left( x_c + k(x - x_c), y_c + k(y - y_c) \right)$ where $(x_c, y_c)$ is the center of dilation and $k$ is the scale factor.

Step 3: Apply the Formula to Each Vertex

For Vertex A(2, 2):

$A'(x'_1, y'_1) = \left( 3 + \frac{1}{2}(2 - 3), -2 + \frac{1}{2}(2 + 2) \right)$ $A'(x'_1, y'_1) = \left( 3 + \frac{1}{2}(-1), -2 + \frac{1}{2}(4) \right)$ $A'(x'_1, y'_1) = \left( 3 - \frac{1}{2}, -2 + 2 \right)$ $A'(x'_1, y'_1) = \left( 2.5, 0 \right)$

For Vertex B(4, 4):

$B'(x'_2, y'_2) = \left( 3 + \frac{1}{2}(4 - 3), -2 + \frac{1}{2}(4 + 2) \right)$ $B'(x'_2, y'_2) = \left( 3 + \frac{1}{2}(1), -2 + \frac{1}{2}(6) \right)$ $B'(x'_2, y'_2) = \left( 3 + 0.5, -2 + 3 \right)$ $B'(x'_2, y'_2) = \left( 3.5, 1 \right)$

For Vertex C(6, 1):

$C'(x'_3, y'_3) = \left( 3 + \frac{1}{2}(6 - 3), -2 + \frac{1}{2}(1 + 2) \right)$ $C'(x'_3, y'_3) = \left( 3 + \frac{1}{2}(3), -2 + \frac{1}{2}(3) \right)$ $C'(x'_3, y'_3) = \left( 3 + 1.5, -2 + 1.5 \right)$ $C'(x'_3, y'_3) = \left( 4.5, -0.5 \right)$

Step 4: Plot the Dilated Points

Now, we have the coordinates of the dilated triangle:

• $A'(2.5, 0)$
• $B'(3.5, 1)$
• $C'(4.5, -0.5)$

These are the new vertices of the triangle after dilation by a scale factor of $\frac{1}{2}$ with the center at $(3, -2)$.

If you plot these points, you'll see that the triangle is smaller and closer to the center of dilation compared to the original triangle.

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Relative Questions:

1. How does the distance of each vertex from the center of dilation change after applying the scale factor?
2. What would be the coordinates if the scale factor was $\frac{3}{2}$ instead of $\frac{1}{2}$?
3. Can you identify situations where dilation might be used in real-world applications?
4. How would you describe the transformation if the scale factor was negative?
5. What other geometric transformations can be combined with dilation to create more complex shapes?

Tip: When dilating, keep track of the changes in both x and y coordinates separately to avoid mistakes in your calculations.