Question

to create pentagon abcde.(x,y)→(\frac{2}{3}x,\frac{2}{3}y)which statement is true?\square each side length of pentagon abcde will be \frac{2}{3} of the corresponding side length of pentagon abcde.\square each angle measure of pentagon abcde will be \frac{2}{3} more than the corresponding angle measure of pentagon abcde.\square each angle measure of pentagon abcde will be \frac{2}{3} of the corresponding angle measure of pentagon abcde.\square each side length of pentagon abcde will be \frac{2}{3} more than the corresponding side length of pentagon abcde.

Explanation:

The transformation given is a dilation with a scale factor of \(\frac{2}{3}\) (since \((x,y)\to(\frac{2}{3}x,\frac{2}{3}y)\)). Dilation affects side lengths (scaling them by the scale factor) but does not change angle measures (angles are preserved in similar figures).

• For side lengths: When a figure is dilated by a scale factor \(k\), each side length of the image is \(k\) times the corresponding side length of the original. So, each side length of pentagon \(A'B'C'D'E'\) should be \(\frac{2}{3}\) of the corresponding side length of pentagon \(ABCDE\), which matches the first statement.
• For angle measures: Dilation preserves angle measures, so angle measures of the image pentagon are equal to the corresponding angle measures of the original, not multiplied by \(\frac{2}{3}\) or increased by \(\frac{2}{3}\). So the statements about angle measures being \(\frac{2}{3}\) of or \(\frac{2}{3}\) more than the original are incorrect. The statement about side lengths being \(\frac{2}{3}\) more is also incorrect (it's a multiple, not an addition).

Answer:

The true statement is: Each side length of pentagon \(A'B'C'D'E'\) will be \(\boldsymbol{\frac{2}{3}}\) of the corresponding side length of pentagon \(ABCDE\) (the first option among the given choices, likely labeled as the first checkbox option with that description).