CircleHardBloom L3
Question
Points $P$ and $Q$ are 3 units apart. $A$ circle centre at $P$ with a radius of 3 units intersects a circle centre $d$ at $Q$ with a radius $\sqrt{3}$ units at point $A$ and $B$. The area of the quadrilateral $APBQ$ is:
Options
A.$\sqrt{99}$
B.$\frac{\sqrt{99}}{2}$
C.$\sqrt{\frac{99}{2}}$
D.$\sqrt{\frac{99}{16}}$
Solution
[Key Concept: Two points $3$ apart with radius-$3$ circles: the intersection geometry gives the required length.]
Area of $APBQ = 2$ (area $\bigtriangleup APQ$ )
$$\begin{matrix} & \ = 2\left( \frac{1}{2} \times \sqrt{3} \times \sqrt{3^{2} - \left( \frac{\sqrt{3}}{2} \right)^{2}} \right) \\ & \ = \sqrt{3} \cdot \frac{\sqrt{33}}{2} = \frac{\sqrt{99}}{2} \end{matrix}$$
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