This question was uploaded on 11/09/21 on social media accounts.
A semicircle is inscribed in a quarter circle such that the semicircle touches both sides of the quarter circle and its edges are on the quarter circle. Find the tangent of the angle between the edge of the quarter circle and the line joining the center of the quarter circle and the end of the diameter of the semicircle.
Solution:
\[tan(45-\alpha)=\frac{r}{r\sqrt2}\]
\[\Rightarrow tan(45-\alpha)=\frac{1}{\sqrt2}\]
Expand tan(45-α)
\[\Rightarrow \frac{tan(45)-tan(\alpha)}{1+tan(45).tan(\alpha)}=\frac{1}{\sqrt2}\]
\[\Rightarrow \frac{1-tan(\alpha)}{1+tan(\alpha)}=\frac{1}{\sqrt2}\]
\[\Rightarrow (\sqrt2+1)tan(\alpha)=\sqrt2-1\]
\[\Rightarrow tan(\alpha)=\frac{\sqrt2-1}{\sqrt2+1}\]
\[\Rightarrow tan(\alpha)=3-2\sqrt2\]
Puzzle related to Geometry, Circle, Angle
