To determine what fraction of polygon [tex]\( Y \)[/tex]'s area is compared to polygon [tex]\( X \)[/tex]'s area, given that polygon [tex]\( Y \)[/tex] is a scaled copy of polygon [tex]\( X \)[/tex] with a scale factor of [tex]\(\frac{1}{3}\)[/tex], we need to understand how the areas of scaled shapes are related to their scale factors.
When a two-dimensional shape is scaled by a factor, the area of the new shape changes by the square of the scale factor. This is because area is a two-dimensional measurement (length × width), and both dimensions are scaled.
Here, the scale factor is [tex]\(\frac{1}{3}\)[/tex]. To find out the changing factor for the area, we square the scale factor:
[tex]\[
\left(\frac{1}{3}\right) ^ 2 = \frac{1}{9}
\][/tex]
Thus, the area of Polygon [tex]\( Y \)[/tex] is [tex]\(\frac{1}{9}\)[/tex] of the area of Polygon [tex]\( X \)[/tex].
This means Polygon [tex]\( Y \)[/tex]'s area is [tex]\(\frac{1}{9}\)[/tex] or approximately 0.1111111111111111 as a decimal fraction of Polygon [tex]\( X \)[/tex]'s area.
