Question

A circle is circumscribed about an equilateral triangle, and that equilateral triangle is circumscribed about another circle. Then the ratio of perimeter of circumcircle to that of an incircle is
A. 3 : 4B. 1 : 2C. 2 : 1D. 4 : 3​

Answer 1

Answer:

Option: C is the answer

Option: C is the answer Explanation:

Let the side of an equilateral triangle is 'a'.

Then radius of a circumcircle is a/√3 and radius of an incircle is a/2√3.

The ratio of perimeter is a/√3 : a/2√3 = 2 : 1

Answer 2

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let the side of the triangle is =a

the midean of the triangle is =(√3/2)a

the radius of the outer circle is

$= \sqrt{ \frac{a {}^{2} }{4} + \frac{a {}^{2} }{12} } \\ = \sqrt{ \frac{4a {}^{2} }{12} } \\ = \sqrt{ \frac{a {}^{2} }{3} } \\ = \frac{a}{ \sqrt{3} }$

the radius of the inner circle is

$= \frac{a}{2 \sqrt{3} }$

therefore the ratio of the circumference will be equal to the ratio of their radius ...

$\frac{ \frac{a}{ \sqrt{3} } }{ \frac{a}{2 \sqrt{ 3} } } \\ = \frac{2}{1}$

=2:1

option (c)2:1

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