Question : The perimeter of an equilateral triangle is equal to the circumference of a circle. The ratio of their areas is: ( Use $\pi =\frac{22}{7}$)
Option 1: $22 : 21\sqrt{3}$
Option 2: $21: 22\sqrt{3}$
Option 3: $21: 22\sqrt{2}$
Option 4: $22: 21\sqrt{2}$
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Correct Answer: $22 : 21\sqrt{3}$
Solution :
Let the side of the equilateral triangle be $a$ unit.
Perimeter = $3a$ unit
Circumference of the circle = $2\pi r$
Given: Perimeter of triangle = circumference of the circle
So, $3a = 2\pi r$
⇒ $\frac{a}{r}=\frac{2 \pi}{3} =\frac{2\times22}{3 \times7}$
$\therefore \frac{a}{r}=\frac{44}{21}$
Area of the equilateral triangle : Area of the circle
= $\frac{\sqrt3}{4}a^2 : \pi r^2$
=$\frac{\sqrt3}{4}\times 44 \times 44:\frac{22}{7}\times21 \times 21$
= $22\sqrt3:21 \times 3$
= $22: 21\sqrt3$
$\therefore$ The ratio of the area of triangle to circle is $22: 21\sqrt3$.
Hence, the correct answer is $22: 21\sqrt3$.
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