Multiple choice

In $\Delta$ $ABC$ the sides opposite to angles $A, B, C$ are denoted by $a, b, c$ respectively. If $\cot\dfrac{A}{2}=30,\ \displaystyle \cot\dfrac{B}{2}=50,\ \displaystyle \cot\frac{C}{2}=70$ then put the sides $a,\ b,\ c$ in ascending order

  1. $b,a,c$
  2. $a,b,c$
  3. $c,b,a$
  4. $a,c,b$
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C Correct answer
Explanation

In a triangle, cot(A/2) = sqrt(s(s-a)/(s-b)(s-c)). Larger cotangent values imply smaller angles. Given cot(A/2)=30, cot(B/2)=50, cot(C/2)=70. A/2 > B/2 > C/2, so A > B > C. In any triangle, larger angles are opposite larger sides. Thus a > b > c. Ascending order: c, b, a.

AI explanation

Using the half-angle identity, cot of an angle divided by 2 equals the semi-perimeter minus the opposite side divided by the inradius r. Since r is constant, the values of (s minus a), (s minus b) and (s minus c) are proportional to 30, 50 and 70 respectively. Adding these three expressions gives 3s minus the sum of the sides, which simplifies to s, meaning s equals 150. Solving for the sides yields a equals 120, b equals 100 and c equals 80. Placing the sides in ascending order gives c, b, a.

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