Overlap of two circles through each other’s centres
Overlap of two circles through each other’s centres
IP 2
7 marks
Two circles each have radius \(6\) cm. The centre of each circle lies on the other circle. The circles meet at \(A\) and \(B\), and their centres are \(P\) and \(Q\).
- Explain why triangles \(APQ\) and \(BPQ\) are equilateral, and hence show that \(\angle APB=\dfrac{2\pi}{3}\) radians.[2]
- Find the exact area of the segment of the circle with centre \(P\) cut off by the chord \(AB\) (on the side of \(Q\)).[3]
- Hence find the exact area and the exact perimeter of the region common to both circles.[2]
Solution:
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Answer:(a) All sides are radii of length \(6\); \(\dfrac{\pi}{3}+\dfrac{\pi}{3}\) (b) \((12\pi-9\sqrt{3})\text{ cm}^2\) (c) Area \((24\pi-18\sqrt{3})\text{ cm}^2\), perimeter \(8\pi\) cm
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