Question

The triangle of maximum area that can be inscribed in a given circle of radius ${}^{\prime }{r}^{\prime }$ is:

• A. A right-angle triangle having two of its sides of length $2r$ and $r$
• B. An equilateral triangle of height $\frac{2r}{3}$
• C. Isosceles triangle with base equal to $2r$
• D. An equilateral triangle having each of its side of length $\sqrt{3}r$

Answer 1

The correct option is D An equilateral triangle having each of its side of length $\sqrt{3}r$

Triangle of maximum area that can be inscribed in a circle is an equilateral triangle. Let $△ABC$ be inscribed in circle,
Now, in $△OBD$
$OD=r\cos {60}^{\circ }=\frac{r}{2}$
Height $=AD=\frac{3r}{2}$
Again in $△ABD$
Now sin ${60}^{\circ }=\frac{3r/2}{AB}$
$\Rightarrow AB=\sqrt{3}r$