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:
\(A = \frac{\sqrt{3}}{4} (12\sqrt{3})^2\)
\(A = \frac{\sqrt{3}}{4} (144 \times 3)\)
\(A = \frac{\sqrt{3}}{4} \times 432\)
\(A = 108\sqrt{3} \, \text{sq.m}\)
Similar Questions
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Important Questions from Mensuration
In a circular garden of radius 15 m, a path of 2 m wide has to be made inside the garden at the rate of ₹ 24 per sq. m. The cost of making the path is: (Take π = \(\frac{22}{7}\) )
A hollow cylinder with outer radius 4 cm and height 2 cm is made up of 1 cm thick metal sheet. What is the volume of metal used? (Take π = \(\frac{22}{7}\))
The sides of a triangular park are 60 m, 297 m and 303 m. Its area is equal to the area of a square-shaped garden. What is the perimeter (in m) of the garden?
The length and breadth of a rectangular field are in the ratio 4 : 3. If the cost of cultivating the field at ₹2 per m 2is ₹600, then the length of the field is:
A cylindrical tank has a capacity of 5632 m3. If the diameter of its base is 8 m, what is the depth of the cylindrical tank? (Use π = \(\frac{22}{7}\))
