Polygon ABCD with vertices at A(−4, 6), B(−2, 2), C(4, −2), and D(4, 4) is dilated using a scale factor of 3/5 to create polygon A′B′C′D′. If the dilation is centered at the origin, determine the vertices of polygon A′B′C′D′.

To determine the vertices of polygon \( A'B'C'D' \), which is a dilation of polygon \( ABCD \) with a scale factor of \( \frac{3}{5} \), we will apply the dilation formula to each vertex. The formula for dilation centered at the origin is:

\[
(x', y') = \left( k \cdot x, k \cdot y \right)
\]
where \( k \) is the scale factor, and \( (x, y) \) are the coordinates of the original vertex.

Given that \( k = \frac{3}{5} \), we will apply this formula to each vertex of polygon \( ABCD \).

### Step-by-Step Calculation for Each Vertex:

1. **Vertex \( A(-4, 6) \):**
   \[
   A' = \left( \frac{3}{5} \cdot -4, \frac{3}{5} \cdot 6 \right) = (-2.4, 3.6)
   \]

2. **Vertex \( B(-2, 2) \):**
   \[
   B' = \left( \frac{3}{5} \cdot -2, \frac{3}{5} \cdot 2 \right) = (-1.2, 1.2)
   \]

3. **Vertex \( C(4, -2) \):**
   \[
   C' = \left( \frac{3}{5} \cdot 4, \frac{3}{5} \cdot -2 \right) = (2.4, -1.2)
   \]

4. **Vertex \( D(4, 4) \):**
   \[
   D' = \left( \frac{3}{5} \cdot 4, \frac{3}{5} \cdot 4 \right) = (2.4, 2.4)
   \]

### Final Coordinates of Polygon \( A'B'C'D' \):

- \( A'(-2.4, 3.6) \)
- \( B'(-1.2, 1.2) \)
- \( C'(2.4, -1.2) \)
- \( D'(2.4, 2.4) \)

These are the vertices of polygon \( A'B'C'D' \) after the dilation with a scale factor of \( \frac{3}{5} \) centered at the origin.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. What happens if the dilation is centered at a point other than the origin?
2. How would the scale factor change if you wanted the image to be larger than the original?
3. What is the effect of a negative scale factor in dilation?
4. How do you calculate the area of the dilated polygon?
5. How does dilation affect the slopes of the lines connecting the vertices?

**Tip:** Remember that in a dilation centered at the origin, all points on the figure move along a straight line from the origin according to the scale factor.

Explore how polygon ABCD with vertices A(-4, 6), B(-2, 2), C(4, -2), and D(4, 4) is dilated using a scale factor of 3/5 centered at the origin. Understand the transformation of each vertex and derive the vertices of polygon A'B'C'D'. Suitable for high school geometry students.

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