Question : What is the ratio of inradius and circumradius of an equilateral triangle?
Option 1: 1 : 2
Option 2: 1 : 3
Option 3: 1 : 4
Option 4: 3 : 2
Correct Answer: 1 : 2
Solution :
Let $a$ be the side of the equilateral triangle.
Circumradius = $\frac{abc}{4 \times \text{Area of triangle}}$, where $a,b,c$ are sides
In Case of equilateral triangle, abc = a
3
and area of triangle = $\frac{\sqrt{3}a^{2}}{4}$
$\therefore$ Circumradius = $\frac{a^{3}}{4 \times \frac{\sqrt3a^{2}}{4}}=\frac{a}{\sqrt{3}}$
Inradius = $\frac{\text{Area of triangle}}{\text{Semi perimeter of triangle}}$
Semi-perimeter = half of the sum of all sides
$\therefore$ Inradius = $\frac{\frac{\sqrt{3}a^{2}}{4}}{\frac{3a}{2}} = \frac{a}{2\sqrt{3}}$
Therefore, Inradius : Circumradius = $\frac{a}{2\sqrt{3}}$: $\frac{a}{\sqrt{3}}$= 1 : 2
Hence, the correct answer is 1 : 2.
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