Trigonometry
Trigonometry
Allen Star Batch
Grade 11
Question:
In a $\triangle ABC$; inscribed circle with centre $I$ touches sides $AB, AC$ and $BC$ at $D, E, F$ respectively. Let area of quadrilateral $ADIE$ is $5$ square units and area of quadrilateral $BFID$ is $10$ square units. Find the value of $$\frac{\cos\left(\frac{C}{2}\right)}{\sin\left(\frac{A-B}{2}\right)}$$
Step-by-Step Solution
Key Concept: Relate the ratio of perimeter segments to inradius and angle properties using half-angle formulas.
From the given conditions $\frac{1}{2}r(AD + AE) = 5$ and $\frac{1}{2}r(BF + BD) = 10$, we derive $\frac{BF + BD}{AD + AE} = 2$. Using the angle bisector property and applying the cosine rule with $\cos\frac{C}{2}$ and $\sin\frac{A-B}{2}$, we get $\frac{\cos\frac{C}{2}}{\sin\frac{A-B}{2}} = 3$.
Correct Answer: 3