Question

The area of an equilateral triangle is $49\sqrt{3}c{m}^2$. Taking each angular point as centre, a circle is described with radius equal to half the length of the side of the triangle. Find the area of the triangle not included in the circle. [ 4 MARKS ]

Answer 1

Diagram: 1 Mark
Concept : 1 Mark
Solution: 2 Marks




$Area=49\sqrt{3}c{m}^2$

Area of an equilateral triangle is given by $\frac{\sqrt{3}}{4}{a}^2$

$\Rightarrow \frac{\sqrt{3}}{4}{a}^2=49\sqrt{3}$

$\Rightarrow a^2=49\times 4\Rightarrow a=14cm$

Thus, radius of each circle is 7 cm

Now, Required area

$=Areaof\Delta ABC-3\times$ (Area of a sector of angle ${60}^{\circ }$ in a circle of radius 7 cm)

$=[49\sqrt{3}-3(\frac{60}{360}\times \frac{22}{7}\times 7^2)]c{m}^2$

$=[49\sqrt{3}-77]c{m}^2=[49\times 1.73-77]c{m}^2=7.77c{m}^2.$