Question

In \(△ABC\), with usual notations, if the sides \(a\), \(b\) and \(c\) are in the ratio \(18:17:7\), then \(cot\frac{A}{2}:cot\frac{B}{2}:cot\frac{C}{2} =\)

• A. \(4:3:14\)
• B. \(14:4:3\)
• C. \(3:4:14\)
• D. \(4:14:3\)

Hint:

Use cot(A/2) = (s - a)/r and find s - a, s - b, s - c.

Answer 1

The Correct Option is C

Step 1: Tangent Half-Angle Idea:
Since $\tan\dfrac A2=\dfrac{r}{s-a}$, the cotangent is $\dfrac{s-a}{r}$.

Step 2: Numbers:
With $s=21$ (taking $k=1$): $s-a=3,\ s-b=4,\ s-c=14$.

Step 3: Answer:
The common factor $1/r$ cancels in the ratio, so it is $3:4:14$. Option (C).

Final Answer:
Option (C). \[ \boxed{\text{(C) } 3:4:14} \]