Question : If the square of the product of three sides of the triangle is 1024 units and the area of its circumcircle is $16 \pi$, what is the area of the given triangle?
Option 1: 2.5 sq. units
Option 2: 3 sq. units
Option 3: 2 sq. units
Option 4: 4 sq. units
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Correct Answer: 2 sq. units
Solution : Area of circle = $\pi R^2$ where $R$ is the radius of the circumcircle. ⇒ $16\pi = \pi R^2$ ⇒ $R = 4$ The length of the three sides is $a,b$, and $c$. The area of the triangle is denoted by $A$. $(abc)^2 = 1024$ ⇒ $(abc) = 32$ Using $R = \frac{abc}{4A}$ ⇒ $4 = \frac{32}{4A}$ ⇒ $A = 2$ Hence, the correct answer is 2 sq. units.
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