4

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. How can I get the area of the region QXROQ bounded by the two semicircular arcs?

mathlover
  • 1,933

2 Answers2

3

This diagram will help:

enter image description here

Note that I have connected $OQ$ and $OR$ - these are both equal to the radius of the blue circle centered at $O$ and so their lengths must be equal to $2$.

I have also labelled the intersection of $QR$ and $OX$ as $Y$. From the question statement it is clear that:$$QY=YR=OY$$ and I have labelled this length as $a$.

Now note in triangle $OYR$ that $OY=YR$ and angle $OYR=90^{\circ}$. This implies that angles $YOR=YRO=45^{\circ}$.

Using similar arguments you can deduce that angles $YOQ=YQO=45^{\circ}$.

This should give you enough information to be able to calculate the value of $a$ and hence calculate the area you require.

Mufasa
  • 5,434
1

Let $S$ be the intersection point between $OX$ and $QR$. Also, let $QS=r$.

I think you can get the area you want using the followings :

  • $QS=SR=SO=r$.

  • $\triangle{SQO}$ and $\triangle{SRO}$ are both isosceles right triangles.

  • $\angle{QOR}=\angle{SOQ}+\angle{SOR}=\frac{\pi}{4}+\frac{\pi}{4}=\frac{\pi}{2}$.

  • The area we want is given by $[\text{semicircle $QRO$}]+[\text{figure $QXR$}]$.

  • $[\text{figure $QXR$}]=[\text{sector $OQXR$}]-[\triangle{OQR}]$.

mathlove
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