To solve the problem of finding the area of an equilateral triangle given the perpendicular distances from an interior point to each of its sides, we can follow these steps:
1. Identify the given distances:
- The perpendicular distances from the interior point to each side are:
[tex]\[ s_1 = 5 \text{ cm}, \quad s_2 = 6 \text{ cm}, \quad s_3 = 9 \text{ cm} \][/tex]
2. Calculate the height of the triangle:
- In an equilateral triangle, the sum of the perpendicular distances from any interior point to the three sides equals the height ( [tex]\(h\)[/tex] ) of the triangle.
[tex]\[ h = s_1 + s_2 + s_3 \][/tex]
[tex]\[ h = 5 \text{ cm} + 6 \text{ cm} + 9 \text{ cm} = 20 \text{ cm} \][/tex]
3. Determine the side length of the triangle:
- The height of an equilateral triangle is given by the formula:
[tex]\[ h = \left(\frac{\sqrt{3}}{2}\right) \times \text{side length} \][/tex]
- Rearrange this formula to solve for the side length ([tex]\(a\)[/tex]):
[tex]\[ \text{side length} = \frac{2h}{\sqrt{3}} \][/tex]
- Substitute [tex]\(h = 20 \text{ cm}\)[/tex]:
[tex]\[ \text{side length} = \frac{2 \times 20 \text{ cm}}{\sqrt{3}} = \frac{40 \text{ cm}}{\sqrt{3}} \approx 23.094 \text{ cm} \][/tex]
4. Calculate the area of the equilateral triangle:
- The area of an equilateral triangle is given by the formula:
[tex]\[ A = \left(\frac{\sqrt{3}}{4}\right) \times (\text{side length})^2 \][/tex]
- Substitute the side length:
[tex]\[ A = \left(\frac{\sqrt{3}}{4}\right) \times \left(\frac{40}{\sqrt{3}}\right)^2 \][/tex]
[tex]\[ A = \left(\frac{\sqrt{3}}{4}\right) \times (23.094)^2 \approx 230.94 \text{ cm}^2 \][/tex]
Therefore, the area of the equilateral triangle is approximately [tex]\(230.94 \text{ cm}^2\)[/tex].
