UPSC CDS Math Area and Perimeter Questions

Consider an equilateral triangle ABC having sides a and a median AD of length l unit'.
In an equilateral triangle, the median is always the perpendicular bisector of the triangle. So,BD=a∕2
In triangle ABD, by pythagoras theorem, we have AB2=AD2+BD2
⟹a2=l2+(‌

a

2

)2
⟹a2=l2+‌

a2

4

⟹‌

3a2

4

=l2 or, ‌‌a2=‌

4l2

3

Now, area of equilateral triangle =‌

√3a2

4

=‌

√3

4

×‌

4l2

3

=‌

√3l2

3