Question

The side of an equilateral triangle is equal to the radius of a circle whose area is 154 cm². The area of the triangle is
(a) 49 cm²
(b) $\frac{49\sqrt{3}}{4}{\mathrm{cm}}^2$
(c) $\frac{7\sqrt{3}}{4}{\mathrm{cm}}^2$
(d) 77 cm²

Answer 1

(b) $\frac{49\sqrt{3}}{4}{\mathrm{cm}}^2$

Area of a circle$=\mathrm{\pi }{r}^2$
$$\begin{gathered} \Rightarrow 154=\mathrm{\pi }{r}^2 \\ \Rightarrow r=\sqrt{\frac{154\times 7}{22}} \\ =\sqrt{7\times 7} \\ =7\mathrm{cm} \end{gathered}$$

The radius of the circle is equal to the side of the equilateral triangle.

∴ r = a (Here, a is the side of the equilateral triangle.)

$a=7\mathrm{cm}$

∴ Area of the equilateral triangle$=\frac{\sqrt{3}}{4}{a}^2=\frac{\sqrt{3}}{4}\times 7\times 7=\frac{49\sqrt{3}}{4}{\mathrm{cm}}^2$