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KVPY2019MathematicsProperties of Triangles

Let A   B   C be an equilateral triangle with side length a. Let R and r denote the radii of the circumcircle and the incircle of triangle A   B   C respectively. Then, as a function of α the ratio R r

Options

  1. Astrictly increases
  2. Bstrictly decreases
  3. Cremains constant
  4. Dstrictly increases for a < 1 and strictly decrease for a > 1

Correct answer

C. remains constant

Step-by-step solution

For an equilateral triangle A B C having side length a. If R and r are radii of the circumcircle and the incircle of triangle A B C respectively, then R = a 2 sec 30 ° = a 2 2 3 = a 3 and r = a 2 tan 30 ° = a 2 × 1 3 = a 2 3 ∴ R r = a 3 a 2 3 = 2 which is independent of a and it is constant.

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