Question

In a triangle ABC, with usual notations, \[ \tan \left( \frac{A}{4} \right) = \frac{5}{6}, \quad \tan \left( \frac{C}{2} \right) = \frac{2}{5}, \] then

• A. \( a, c, b \) are in A.P.
• B. \( b, a, c \) are in A.P.
• C. \( a, b, c \) are in A.P.
• D. \( a, b, c \) are in G.P.

Hint:

When solving geometry problems involving trigonometric identities, look for common relationships like A.P. or G.P. that might simplify the expression.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to use the relation between half-angle tangents and the sides of the triangle.
Step 2: Key Formula or Approach:
\( \tan(A/2) \tan(C/2) = \frac{s - b}{s} \).
Step 3: Detailed Explanation:
Given \( \tan(A/2) = 5/6 \) and \( \tan(C/2) = 2/5 \).
Product: \( \frac{5}{6} \times \frac{2}{5} = \frac{1}{3} \).
Substitute formula: \( \frac{s - b}{s} = \frac{1}{3} \).
\( 3(s - b) = s \Rightarrow 3s - 3b = s \Rightarrow 2s = 3b \).
Since \( 2s = \text{a} + \text{b} + \text{c} \):
\( \text{a} + \text{b} + \text{c} = 3b \).
\( \text{a} + \text{c} = 2b \).
This is the condition for \( \text{a, b, c} \) to be in Arithmetic Progression (A.P.).
Step 4: Final Answer:
\( \text{a, b, c} \) are in A.P.