Show that the triangle of maximum area that can be inscribed in a given circle is an equilateral triangle.

Options

A. `(3a^(2))/(4)`
B. `(a^(2))/(2)`
C. `(3sqrt(3)a^(2))/(4)`
D. `(sqrt(3)a^(2))/(4)`

Answer

C

Explanation

For area of triangle to be maximum, it should be equilateral triangle.
Area of `DeltaOAB = 1/2 ab sin theta`

`1/2.a.a.sin theta`
`= 1/2 a^(2) .sin120^(@)`
`= (sqrt(3))/(4) a^(2)`
Area of triangle `OAB,OBC,OAC`
`=` Area of triangle ABC
`= 3 xx(sqrt(3))/(4)a^(2) = (3sqrt(3))/(4)a^(2)`