To find the area of Polygon [tex]\( D \)[/tex], which is a scaled version of Polygon [tex]\( C \)[/tex] using a scale factor of [tex]\(\frac{1}{2}\)[/tex], follow these steps:
1. Identify the given values:
- The area of Polygon [tex]\( C \)[/tex] is 40 square units.
- The scale factor used to create Polygon [tex]\( D \)[/tex] from Polygon [tex]\( C \)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
2. Understand the relationship between the areas of similar polygons:
- When a polygon is scaled by a certain factor, the area of the new polygon is the area of the original polygon multiplied by the square of the scale factor. This is because the area is a two-dimensional measure, and scaling affects both dimensions.
3. Calculate the area of Polygon [tex]\( D \)[/tex]:
- The scale factor is [tex]\(\frac{1}{2}\)[/tex]. Therefore, you need to square the scale factor to determine the factor by which the area changes.
[tex]\[
\left( \frac{1}{2} \right)^2 = \frac{1}{4}
\][/tex]
- Multiply the area of Polygon [tex]\( C \)[/tex] by [tex]\(\frac{1}{4}\)[/tex] to find the area of Polygon [tex]\( D \)[/tex]:
[tex]\[
\text{Area of Polygon } D = 40 \text{ square units} \times \frac{1}{4} = 10 \text{ square units}
\][/tex]
Thus, the area of Polygon [tex]\( D \)[/tex] is [tex]\( 10 \)[/tex] square units.
