Question

$\Delta ABC$ is an isosceles right triangle with area $P$. The radius of the circle that passes through the point $A,B$ and $C$ is:

• A. $\sqrt{P}$
• B. $\sqrt{\frac{P}{2}}$
• C. $\frac{\sqrt{P}}{2}$
• D. $\sqrt{2P}$

Answer 1

The correct option is A $\sqrt{P}$

$△ABC$ is a right angled triangle.
A circle passes through $A,B$ and $C$.
$\therefore AB$ is the diameter of the circle and $\angle ACB={90}^o$
$AC=BC=x$
Let radius of the circle be $r$
$A{B}^2=A{C}^2+B{C}^2$
$\Rightarrow (2r{)}^2=x^2+x^2$
$\Rightarrow 4r^2=2x^2$
$\Rightarrow r^2=\frac{x^2}{2}$
Also,
Area of $△ABC=\frac{1}{2}\times AC\times BC$
$\Rightarrow P=\frac{1}{2}\times x^2$
$\therefore r^2=P$
$\Rightarrow r=\sqrt{p}$