To determine the fraction of Polygon [tex]$X$[/tex]'s area that Polygon [tex]$Y$[/tex]'s area represents, we'll start with the given scale factor. In this case, the scale factor is [tex]\(\frac{1}{3}\)[/tex].
Recall that when dealing with similar polygons, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths. Here, the corresponding side lengths are scaled by a factor of [tex]\(\frac{1}{3}\)[/tex].
To find the ratio of the areas, we need to square this scale factor:
[tex]\[
\left(\frac{1}{3}\right)^2
\][/tex]
Calculating the square of [tex]\(\frac{1}{3}\)[/tex] gives us:
[tex]\[
\left(\frac{1}{3}\right) \times \left(\frac{1}{3}\right) = \frac{1}{9}
\][/tex]
Thus, the area of Polygon [tex]$Y$[/tex] is [tex]\(\frac{1}{9}\)[/tex] of the area of Polygon [tex]$X$[/tex].
Expressed as a decimal, this fraction is:
[tex]\[
\frac{1}{9} = 0.1111111111111111
\][/tex]
Therefore, Polygon [tex]$Y$[/tex]'s area is [tex]\(\frac{1}{9}\)[/tex], or approximately [tex]\(0.1111\)[/tex], of Polygon [tex]$X$[/tex]'s area.
