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If the length of median of an equilateral triangle be $x,$ then its area is

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x 2

32 x 2

x 2 3

x 2 2

answer is C.

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Detailed Solution

We have been given an equilateral triangle ABC such that the length of median is

x .

∴ A D = x

We know that the median is also an altitude in the equilateral triangle.
Let

a

be the side of the triangle. So,

A B = B C = A C = a

Since AD is the median of

Δ ABC,

therefore, we have,

⇒ BD = a 2 cm

In right

Δ ADB,

we have,
AB

2 = B D 2 + A D 2

⇒ a 2 = a 2 2 + x 2

⇒ x 2 = 3 a 2 4

⇒ x = 32 a ⇒ a = 2 x 3

We know that,

Area of Δ A B C = 12 × b × h ⇒ Area of Δ A B C = 12 × 2 x 3 × x ⇒ Area of Δ A B C = x 2 3

Therefore, if the length of median of an equilateral triangle be

x

, then its area is

x 2 3 .

Hence, option (3) is correct.

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