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MAT Exam 2022 Paper

Question : 100 of 200

Marks: +1, -0

The lengths of perpendiculars drawn from any point inside a equilateral triangle a1, a2 and a3 respectively. Find the length of the side of equilateral triangle?

$\frac{2}{\sqrt{3}}\left(\mathrm{a}_{1}+\mathrm{a}_{2}+\mathrm{a}_{3}\right)$
$\frac{1}{3}\left(a_{1}+a_{2}+a_{3}\right)$
$\frac{1}{\sqrt{3}}\left(\mathrm{a}_{1}+\mathrm{a}_{2}+\mathrm{a}_{3}\right)$
$\frac{4}{\sqrt{3}}\left(\mathrm{a}_{1}+\mathrm{a}_{2}+\mathrm{a}_{3}\right)$

Validate

Solution:

Concept:

In an equilateral triangle, the sum of the perpendicular distances from any interior point to the three sides is constant and equals the altitude of the triangle.

Explanation:

Let the side of the equilateral triangle be

x

Its altitude is

h = \frac{\sqrt{3}}{2}x

For any interior point, the sum of the perpendiculars is equal to the altitude:

a_1 + a_2 + a_3 = h

Substitute the altitude:

a_1 + a_2 + a_3 = \frac{\sqrt{3}}{2}x

Solve for

x

x = \frac{2}{\sqrt{3}}(a_1 + a_2 + a_3)

Thus, the side length depends only on the sum of the three perpendicular distances.

Answer:

\frac{2}{\sqrt{3}}\left(a_1 + a_2 + a_3\right)

So the correct option is A.

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