Hi guys. I need a lot of help on this.

"Equilateral triangle ABC has centroid G. Triangle A'B'C' is the image of triangle ABC upon a dilation with center G and scale factor -2/3. Let K be the area of the region that is within both triangles. Find K/[ABC]."
How do you get a negative scale factor?

a negative scale factor changes size, and reflects through the dilation center.

If it were positive, the whole reduced triangle would of course remain within ABC. But, by flipping through G as well as scaling, some of A'B'C' will lie outside of ABC.

So, did you get 11/27?

To understand how we get a negative scale factor, let's first review what a scale factor represents.

In a dilation, the scale factor determines how the size of the original figure changes in relation to the dilated figure.

A positive scale factor greater than 1 will make the dilated figure larger than the original figure, while a positive scale factor between 0 and 1 will make the dilated figure smaller than the original figure.

On the other hand, a negative scale factor will not only change the size of the figure but also flip it across the center of dilation. This means that the dilated figure will be both smaller and flipped compared to the original figure.

To understand how a negative scale factor works, let's first review what a scale factor represents in a dilation.

A dilation is a transformation that changes the size of a shape without altering its shape or orientation. The scale factor determines how much the shape is enlarged or reduced.

A positive scale factor greater than 1 enlarges the shape, while a positive scale factor between 0 and 1 reduces the shape.

On the other hand, a negative scale factor flips or reflects the shape about the center of dilation before enlarging or reducing it. This transformation can be visualized as a mirror image of the shape.

In the given problem, the scale factor is -2/3, which means that after reflecting the shape about the center of dilation, it will be reduced by a factor of 2/3.