If y is the length of the median of an equilateral triangle, then find its area.

Correct option is C

Given:

y is the length of the median of an equilateral triangle.

Formula used:

Area of an equilateral triangle =$$\frac{\sqrt{3}}{4} \times \text{side}^2$$​​

Relation between the median (y) and the side (s) of an equilateral triangle:

y = $$\left(\frac{\sqrt{3}}{2}\right) \times s$$​

Solution:

From the relation between median and side:

​$$y = \left(\frac{\sqrt{3}}{2}\right) \times s \implies s = \frac{2y}{\sqrt{3}}$$

​$$\text{Area} = \left(\frac{\sqrt{3}}{4}\right) \times s^2$$​

Substituting s =$$\frac{2y}{\sqrt{3}}$$​ gives:

Area=$$\left(\frac{\sqrt{3}}{4}\right) \times \left(\frac{2y}{\sqrt{3}}\right)^2$$​

Area= $$\left(\frac{\sqrt{3}}{4}\right) \times \left(\frac{4y^2}{3}\right)$$​

Area $$= \frac{\sqrt{3} \times 4y^2}{3 \times 4} = \frac{y^2}{\sqrt{3}}$$​