This question was uploaded on 10/09/21 on social media accounts.
Find the area of the semicircle inscribed in a quarter circle such that the semicircle touches both sides of the quarter circle and its edges are on the quarter circle. The distance between the touching point of the semicircle and the edge of the quarter circle is 2.
Solution:
\[(r+2)^2= \left(r\sqrt{2}\right)^2+r^2\]
\[\Rightarrow r^2+4r+4= 3r^2\]\[\Rightarrow 2r^2-4r-4=0\]\
\[\Rightarrow r^2-2r-2=0\]
\[\Rightarrow r=\frac{2\pm\sqrt{4+8}}{2}\]
\[\Rightarrow r=\frac{2\pm2\sqrt{3}}{2}\]
\[\Rightarrow r=1\pm\sqrt{3}\]
Here r is positive,
So we will skip negative value:
\[\Rightarrow r=1+\sqrt{3}\]
Now,
\[Area=\frac{\pi(1+\sqrt{3})^2}{2}\]
\[\Rightarrow Area=\frac{\pi(4+2\sqrt{3})}{2}\]
\[\Rightarrow Area=\pi(2+\sqrt{3})\]
Puzzle based on Geometry, Circle, Area
