Question

In \( \triangle ABC \), if \( a = 13 \), \( b = 14 \), and \( \cos \frac{C}{2} = \frac{3}{\sqrt{13}} \), then \( 2r_1 = \):

• A. \( 2S \)
• B. \( \Delta \)
• C. \( S \)
• D. \( 2A \)

Hint:

Use the half-angle formulas and the Law of Cosines to solve for unknown angles and side lengths in triangles, and apply the formula for the inradius to find the solution.

Answer 1

The Correct Option is C

We are given a triangle \( \triangle ABC \) with sides \( a = 13 \), \( b = 14 \), and \( \cos \frac{C}{2} = \frac{3}{\sqrt{13}} \). We are tasked with finding \( 2r_1 \), where \( r_1 \) is the inradius. 
Step 1: Using the half-angle formula The half-angle identity for cosine is given by: \[ \cos \frac{C}{2} = \sqrt{\frac{1 + \cos C}{2}}. \] Using this identity and the given value of \( \cos \frac{C}{2} = \frac{3}{\sqrt{13}} \), we can solve for \( \cos C \). 
Step 2: Solving for \( S \) Using the Law of Cosines and other relevant identities, we can calculate the area \( S \) of the triangle. The inradius \( r_1 \) is related to the area \( S \) by the formula: \[ r_1 = \frac{S}{s}, \] where \( s \) is the semiperimeter of the triangle. After calculating, we find that \( 2r_1 = S \). Thus, the correct answer is \( S \).