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Q.

Let ABC be an equilateral triangle with side length a. Let R and r denote the radii of the circumcircle and the incircle of triangle ABC respectively. Then, as a function of a, the ratio $R r$ is:

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a

Strictly increases

b

Strictly decreases

c

Remains constant

d

Strictly increases for

a < 1

and strictly decreases for

a > 1

answer is C.

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Detailed Solution

Let the diagram of the situation be,
The equilateral triangle has an angle of

60 ∘

. We know that

sin 60 ∘ = 32

. So, in

Δ C A F

, we have

sin A = C F A C sin 60 ∘ = C F a 32 = C F a C F = 32 a

From the figure, we have

C F = C D + D F

. We now have the property that the centroid of an equilateral triangle coincides with both the circumcircle and the incircle. We also know that the centroid of an equilateral triangle divides the median in the ratio 1:2, and that the median, altitude, and angle bisector drawn from the same vertex coincide in an equilateral triangle.

C D = 23 C F C D = 23 × 32 a C D = 13 a

In the similar way, we have

D F = 13 C F D F = 13 × 32 a = 123 a

Then, the ratio is,

R r = 13 a 123 a = 233 = 2

As a result, the ratio of the radii is constant and does not depend on the side length of the equilateral triangle.
Hence, the correct option is 3.

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Let ABC be an equilateral triangle with side length a. Let R and r denote the radii of the circumcircle and the incircle of triangle ABC respectively. Then, as a function of a, the ratio R r is: