Question

which statements about finding the area of the equilateral triangle are true? select three options. the apothem can be found using the pythagorean theorem. the apothem can be found using the tangent ratio. the perimeter of the equilateral triangle is 15 cm. the length of the apothem is approximately 2.5 cm. the area of the equilateral triangle is approximately 65 cm²

Explanation:

Step1: Recall properties of equilateral - triangle and apothem

In an equilateral triangle, the apothem is the perpendicular distance from the center to a side. If we consider half - of a side and the distance from the vertex to the center (a line segment), we can form a right - triangle. The Pythagorean theorem can be used to find the apothem if we know the side length and the distance from the vertex to the center. So, the apothem can be found using the Pythagorean theorem.

Step2: Analyze tangent ratio for apothem

If we know the central angle and the distance from the vertex to the center, we can use the tangent ratio in the right - triangle formed by the apothem, half - of a side, and the line from the vertex to the center to find the apothem. So, the apothem can be found using the tangent ratio.

Step3: Calculate perimeter

The side length of the equilateral triangle is 8.7 cm. The perimeter \(P\) of an equilateral triangle with side length \(s\) is \(P = 3s\). Here, \(P=3\times8.7 = 26.1\) cm, not 15 cm.

Step4: Calculate apothem

If we assume the right - triangle formed by the apothem \(a\), half - of a side \(b\), and the line from the vertex to the center \(l\). If \(l = 5\) cm and \(b=\frac{8.7}{2}=4.35\) cm, using the Pythagorean theorem \(a=\sqrt{l^{2}-b^{2}}=\sqrt{25 - 18.9225}=\sqrt{6.0775}\approx2.5\) cm.

Step5: Calculate area

The area \(A\) of a regular polygon is \(A=\frac{1}{2}aP\), where \(a\) is the apothem and \(P\) is the perimeter. \(a\approx2.5\) cm and \(P = 26.1\) cm, so \(A=\frac{1}{2}\times2.5\times26.1 = 32.625\) \(cm^{2}\), not approximately \(65\) \(cm^{2}\).

Answer:

The apothem can be found using the Pythagorean theorem.
The apothem can be found using the tangent ratio.
The length of the apothem is approximately 2.5 cm.