Question

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

(D) $\sqrt{2}{r}^2$$sq.units$

Answer 1

The largest triangle that can be inscribed in a semi-circle of radius $r$ units is the triangle having its base as the diameter of the semi-circle and the two other sides are taken by considering a point C on the circumference of the semi-circle and joining it by the end points of diameter A and B.

$\therefore \angle C=90°$ (by the properties of the circle)

So, ΔABC is a right-angled triangle with the base as diameter AB of the circle and height be CD.

Height of the triangle $=r$

∴ Area of largest ΔABC $=$ $\left(\frac{1}{2}\right)$ $\times$Base$\times$ Height

$=$ $\left(\frac{1}{2}\right)$$\times$ AB $\times$ CD

$=\left(\frac{1}{2}\right)\times 2r\times r$

$=r^2$ $sq.units$

Therefore, the Area of the largest ΔABC$=r^2$$sq.units$