This question was uploaded on 28/09/21 on social media accounts.
A semicircle is drawn inside a square on a side of the square. A line is passing through one vertex to the midpoint of the other side of the square.
Solution 1:
Two triangles are similar:
The hypotenuse of a triangle with sides a, 2a:
\[Hypo. = a\sqrt5\]
Similar triangles:
\[\frac{a\sqrt5}{a}=\frac{a}{2}\]
\[\Rightarrow a=\sqrt{20}\]
Area:
\[A = \pi\frac{(\sqrt{20})^2}{4}=10\pi\]
Solution 2:
Two triangles are similar:
The hypotenuse of a triangle with sides a, 2a:
\[Hypo. = a\sqrt5\]
Similar triangles:
\[\frac{a\sqrt5}{a}=\frac{2a}{4}\]
\[\Rightarrow a=\sqrt{20}\]
Area:
\[A = \pi\frac{(\sqrt{20})^2}{4}=10\pi\]
Puzzle related to Geometry, Square, Circle, Area
