To find the area of a regular polygon with a given perimeter [tex]\( P \)[/tex] and apothem length [tex]\( a \)[/tex], you can use a specific geometric formula. Let's consider the options provided and determine which one is correct.
The formula to find the area [tex]\( A \)[/tex] of a regular polygon with perimeter [tex]\( P \)[/tex] and apothem length [tex]\( a \)[/tex] is:
[tex]\[ A = \frac{1}{2} \cdot P \cdot a \][/tex]
This means that the area is half the product of the perimeter and the apothem. Let's examine the options:
A. [tex]\( A = \frac{1}{2} (P \cdot a) \)[/tex]
This option matches our derived formula perfectly. It states that the area [tex]\( A \)[/tex] is half the product of the perimeter [tex]\( P \)[/tex] and the apothem [tex]\( a \)[/tex].
B. [tex]\( a = P \cdot A \)[/tex]
This option is incorrect because it suggests that the apothem [tex]\( a \)[/tex] is the product of the perimeter [tex]\( P \)[/tex] and the area [tex]\( A \)[/tex], which does not correspond to the formula we need.
C. [tex]\( a = \frac{1}{2} (P \cdot A) \)[/tex]
This option is incorrect as it implies the apothem [tex]\( a \)[/tex] is half the product of the perimeter [tex]\( P \)[/tex] and the area [tex]\( A \)[/tex], which is not how we calculate the apothem or the area.
D. [tex]\( A = P \cdot a \)[/tex]
This option is incorrect because it states that the area [tex]\( A \)[/tex] is the product of the perimeter [tex]\( P \)[/tex] and the apothem [tex]\( a \)[/tex] without the factor of [tex]\(\frac{1}{2}\)[/tex], which is crucial in the correct formula.
Given these options, the correct formula is:
[tex]\[ A = \frac{1}{2} (P \cdot a) \][/tex]
Therefore, the correct answer is:
A. [tex]\( A = \frac{1}{2} (P \cdot a) \)[/tex]
