To find the area [tex]\(A\)[/tex] of a regular polygon using its perimeter [tex]\(P\)[/tex] and its apothem length [tex]\(a\)[/tex], you need to follow these steps:
1. Understand what the apothem is: The apothem of a regular polygon is a line from the center to the midpoint of one of its sides and is perpendicular to that side.
2. Recall the formula to calculate the area of a regular polygon: The area [tex]\(A\)[/tex] is given by the formula:
[tex]\[
A = \frac{1}{2} \times P \times a
\][/tex]
In this formula:
- [tex]\(P\)[/tex] is the perimeter of the polygon.
- [tex]\(a\)[/tex] is the length of the apothem.
- [tex]\(\frac{1}{2}\)[/tex] is a constant that is part of the formula.
Given the choices:
A. [tex]\(a = \frac{1}{2}(P A)\)[/tex] is incorrect because it places the apothem [tex]\(a\)[/tex] on the left, which isn't logically in line with the formula we're deriving.
B. [tex]\(A = P a\)[/tex] is incorrect because it omits the [tex]\(\frac{1}{2}\)[/tex] constant that is essential in the correct formula.
C. [tex]\(A = \frac{1}{2}(P a)\)[/tex] is correct because it correctly incorporates the [tex]\(\frac{1}{2}\)[/tex] factor necessary to calculate the area.
D. [tex]\(a = P A\)[/tex] is incorrect because it mistakenly swaps the concepts of area [tex]\(A\)[/tex] and apothem [tex]\(a\)[/tex] and does not work within the context of the correct formula.
Thus, the correct answer is:
C. [tex]\(A = \frac{1}{2}(P a)\)[/tex]
