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.