Question

Question 4
Area of the largest triangle that can be inscribed in a semi – circle of radius r units is:
(A) $r^{2}squnits$
(B) $\frac{1}{2}{r}^{2}squnits$
(C) $2r^{2}squnits$
(D) $\sqrt{2}{r}^{2}squnits$

Answer 1

Take a point C on the circumference of the semi circle and join it by the end points of diameter A and B
$\therefore \angle C={90}^{\circ }$
[angle in a semi-circle is a right angle]
Area of $\Delta ABC$
$=\frac{1}{2}\times \left(base\right)\left(height\right)$

So, $\Delta ABC$ is right angled triangle.

$\therefore Areaoflargest\Delta ABC=\frac{1}{2}\times AB\times CD$
Inorder to have largest traingle, base of triangle will be circles diameter and maximum height can be acheived when the point c is taken just above centre. i.e. CD = radius of circle.

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

$=r^{2}squnits$