Question

If cotangents of half the angles of a triangle are in A.P., then the sides of the triangle are in .

• A. A.P.
• B. G.P.
• C. H.P.
• D. A.G.P.

Answer 1

The correct option is A A.P.

$\mathrm{We}\mathrm{know}\mathrm{that},\cot \frac{A}{2}=\sqrt{\frac{s(s-a)}{(s-b)(s-c)}}=\frac{s(s-a)}{\Delta }\mathrm{Similarly},\cot \frac{B}{2}=\frac{s(s-b)}{\Delta }\mathrm{and}\cot \frac{C}{2}=\frac{s(s-c)}{\Delta }\mathrm{As},\cot \frac{A}{2},\cot \frac{B}{2}\mathrm{and}\cot \frac{C}{2}\mathrm{are}\mathrm{in}A.P.\Rightarrow \frac{s(s-a)}{\Delta },\frac{s(s-b)}{\Delta }\mathrm{and}\frac{s(s-c)}{\Delta }\mathrm{are}\mathrm{in}A.P.\Rightarrow s-a,s-b\mathrm{and}s-c\mathrm{are}\mathrm{in}A.P.\Rightarrow a,b,c\mathrm{will}\mathrm{be}\mathrm{in}A.P.$