Certainly! Let's go through the step-by-step process to solve the given equation for [tex]\( a \)[/tex]:
We start with the given equation for the area of a regular polygon:
[tex]\[ A = \frac{1}{2} P a \][/tex]
Here, [tex]\( A \)[/tex] represents the area, [tex]\( P \)[/tex] represents the perimeter of the polygon, and [tex]\( a \)[/tex] represents the apothem.
### Step 1: Eliminate the fraction
To isolate [tex]\( a \)[/tex], we want to eliminate the fraction [tex]\( \frac{1}{2} \)[/tex]. We can do this by multiplying both sides of the equation by 2:
[tex]\[ 2A = 2 \left( \frac{1}{2} P a \right) \][/tex]
This simplifies to:
[tex]\[ 2A = P a \][/tex]
### Step 2: Solve for [tex]\( a \)[/tex]
Next, we need to isolate [tex]\( a \)[/tex] on one side of the equation. To do this, we will divide both sides of the equation by [tex]\( P \)[/tex]:
[tex]\[ \frac{2A}{P} = a \][/tex]
Thus, we have:
[tex]\[ a = \frac{2A}{P} \][/tex]
### Conclusion
The correct rearranged formula for [tex]\( a \)[/tex] in terms of [tex]\( A \)[/tex] and [tex]\( P \)[/tex] is:
[tex]\[ a = \frac{2A}{P} \][/tex]
This matches one of the given options. The correct choice is:
[tex]\[ a = \frac{2A}{P} \][/tex]
