Question

In \( \triangle ABC \), if \[ \cot \frac{A}{2} : \cot \frac{B}{2} : \cot \frac{C}{2} = 3 : 7 : 9 \] then \( a : b : c = \) ?

• A. \( 8 : 6 : 5 \)
• B. \( 5 : 6 : 8 \)
• C. \( 10 : 8 : 5 \)
• D. \( 5 : 8 : 10 \)

Hint:

When given the cotangent ratio, take the reciprocal to obtain the side ratios of the triangle.

Answer 1

The Correct Option is A

We are given that: \[ \cot \frac{A}{2} : \cot \frac{B}{2} : \cot \frac{C}{2} = 3 : 7 : 9 \] This ratio is related to the sides of the triangle by the formula: \[ \frac{a}{\sin \frac{A}{2}} = \frac{b}{\sin \frac{B}{2}} = \frac{c}{\sin \frac{C}{2}} = 2R \] Thus, the ratio of the sides is the inverse of the cotangent ratio. Therefore, we get: \[ a : b : c = \frac{1}{\cot \frac{A}{2}} : \frac{1}{\cot \frac{B}{2}} : \frac{1}{\cot \frac{C}{2}} = 8 : 6 : 5 \] Hence, the correct answer is option (1) \( 8 : 6 : 5 \).