Trigonometry
Trigonometry
Allen Star Batch
Grade 11
Question:
A quadrilateral $ABCD$ in which $AB = a$, $BC = b$, $CD = c$ and $DA = d$ is such that one circle can be inscribed in it and another circle can be circumscribed about it. $\cos A =$
$\frac{ad + bc}{ad - bc}$
$\frac{ad - bc}{ad + bc}$
$\frac{ac + bd}{ac - bd}$
$\frac{ac - bd}{ac + bd}$
Step-by-Step Solution
Key Concept: Combining the tangential property ($a+c=b+d$) with the cyclic property ($C=\pi-A$) constrains the angles.
For a cyclic quadrilateral with inscribed circle, $a+c=b+d$ and $C=\pi-A$. Using the cosine rule for diagonal $BD$: $2ad\cos A = a^2 + d^2 - (b^2 + c^2 - 2bc\cos C)$. Substitute $\cos C = -\cos A$ and the constraint $a+c=b+d$ to derive $\cos A = \frac{ad-bc}{ad+bc}$.
Correct Answer: 2