Let's go through the necessary steps to answer the question:
1. Calculate the Apothem:
- Given: Side length of the equilateral triangle is 10 units.
- To calculate the apothem [tex]\( a \)[/tex], use the formula:
[tex]\[
a = \frac{\text{side length}}{2 \cdot \tan(30^\circ)}
\][/tex]
- The apothem, rounded to the nearest tenth, is [tex]\( 8.7 \)[/tex] units.
2. Calculate the Perimeter:
- The perimeter [tex]\( P \)[/tex] of an equilateral triangle is calculated as:
[tex]\[
P = 3 \times \text{side length}
\][/tex]
- Therefore, the perimeter is [tex]\( 3 \times 10 = 30 \)[/tex] units.
3. Calculate the Area Using the Formula for a Regular Polygon:
- The area [tex]\( A \)[/tex] of a regular polygon can be calculated using the formula:
[tex]\[
A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\][/tex]
- Plugging in the values:
[tex]\[
A = \frac{1}{2} \times 30 \times 8.7 = 129.9038105676658 \text{ units}^2
\][/tex]
4. Compare to Bianca's Answer:
- Bianca's calculated area was [tex]\( 43.5 \)[/tex] units[tex]\(^2\)[/tex].
So, summarizing the results:
- The apothem, rounded to the nearest tenth, is 8.7 units.
- The perimeter of the equilateral triangle is 30 units.
- Therefore, the area of the equilateral triangle is 129.9038105676658 or approximately 43.5 units[tex]\(^2\)[/tex].
- The calculated areas are:
- Bianca's calculated area: [tex]\( 43.5 \)[/tex] units[tex]\(^2\)[/tex]
- Correctly calculated area: [tex]\( 129.9038105676658 \)[/tex] units[tex]\(^2\)[/tex]
