Question

The area of the largest triangle that can be inscribed in a semi-circle of radius $r$ is
(a) $2r^2$ sq. units
(b) $r^2$ sq. units
(c) $\frac{1}{2}{r}^2$ sq. units
(d) $\sqrt{2}{r}^2$ sq. units

Answer 1

Correct option is (b).

The triangle with the largest area will be symmetrical as shown in the figure.

Let the radius of the circle be $OC=r$ units.

Join $OA$

Now, $OA=r$ units and $OA\perp BC$ [By construction]

Therefore, area of the $\Delta ABC=\frac{1}{2}\times BC\times OA$

$=\frac{1}{2}\times 2r\times r$

$=r^2$ sq. units

Therefore, the required area is $r^2$ sq. units