Question

If square ABCD is dilated by a scale factor of 1/3


with a dilation center at vertex A, by what factor does the area of the square decrease?

Answer 1

When a shape is dilated by a scale factor, the area changes by the square of that scale factor.

In this case, the square ABCD is being dilated by a scale factor of \( \frac{1}{3} \). To find the change in area, we square the scale factor:

\[
\text{New Area Factor} = \left( \frac{1}{3} \right)^2 = \frac{1}{9}
\]

This means that the area of the dilated square will be \( \frac{1}{9} \) of the original area.

To determine how much the area decreases, you can subtract the new area factor from 1:

\[
\text{Decrease in Area} = 1 - \frac{1}{9} = \frac{9}{9} - \frac{1}{9} = \frac{8}{9}
\]

Thus, the area of the square decreases by a factor of \( \frac{8}{9} \).

Therefore, the area of the square decreases by \( \frac{8}{9} \) of its original area after the dilation.