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: Setup:
Let \(a=18k,\ b=17k,\ c=7k\). The semi-perimeter is \(s=\dfrac{18k+17k+7k}{2}=21k\).

Step 2: Key Formula:
In a triangle, \(\cot\dfrac A2=\dfrac{s-a}{r}\), \(\cot\dfrac B2=\dfrac{s-b}{r}\), \(\cot\dfrac C2=\dfrac{s-c}{r}\), where \(r\) is the inradius.

Step 3: Compute:
\(s-a=3k,\ s-b=4k,\ s-c=14k\).

Step 4: Ratio:
\[ \cot\frac A2:\cot\frac B2:\cot\frac C2=(s-a):(s-b):(s-c)=3:4:14 \]
Option (B) \(14:4:3\) and the others reorder the same numbers, so the correct assignment matters: the largest side \(a\) goes with the smallest \(s-a\), giving \(3:4:14\).

Final Answer:
The ratio is \(3:4:14\), option (C). \[ \boxed{\text{(C) } 3:4:14} \]