Equilateral Triangle Altitude Calculation

For an equilateral triangle, the sum of the perpendicular distances from any interior point to its sides is equal to the triangle's altitude (\(h\)).

Given perpendicular lengths are 5 m, 6 m, and 7 m.

Therefore, the altitude \(h\) is:

\(h = 5 \, \text{m} + 6 \, \text{m} + 7 \, \text{m} = 18 \, \text{m}\)

Equilateral Triangle Side Length Derivation

The relationship between the altitude (\(h\)) and the side length (\(s\)) of an equilateral triangle is given by the formula:

\(h = \frac{\sqrt{3}}{2} s\)

Substitute the calculated altitude:

\(18 = \frac{\sqrt{3}}{2} s\)

Solve for \(s\):

\(s = \frac{2 \times 18}{\sqrt{3}} = \frac{36}{\sqrt{3}}\)

Rationalize the denominator:

\(s = \frac{36}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{36\sqrt{3}}{3} = 12\sqrt{3} \, \text{m}\)

Equilateral Triangle Area Calculation

The area (\(A\)) of an equilateral triangle with side length \(s\) is calculated using the formula:

\(A = \frac{\sqrt{3}}{4} s^2\)

Substitute the side length \(s = 12\sqrt{3}\) m:

\(A = \frac{\sqrt{3}}{4} (12\sqrt{3})^2\)

\(A = \frac{\sqrt{3}}{4} (144 \times 3)\)

\(A = \frac{\sqrt{3}}{4} \times 432\)

\(A = 108\sqrt{3} \, \text{sq.m}\)