Find the area…


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    swastik pramanik

    PS is a line segment of length 4 and O is the midpoint of PS. A semicircular arc is drawn with PS as diameter. Let X be the midpoint of this arc. Q and R are points on the arc PXS such that QR is parallel to PS and the semicircular arc drawn with QR as diameter is tangent to PS. What is the area of the region QXROQ bounded by the two semicircular arcs?


    Let us denote \(M=XO\cap QR\). It is clear that the triangle \(OMQ\) is right isosceles and \(OQ=2\). From this information, we can find \(OM\). This in turn gives the area of the bounded arc \(QRO\). Again, we know that \(\angle QOR=\frac{\pi}{2}\)  hence we know the area of the sector bounded by the arc \(QXR\). Subtracting the area of the triangle \(QRO\) we get the area of the region bounded by \(QR\) and the arc \(QXR\). Convince yourself that this is all we need.

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