This question is uploaded on Instagram on page "@mymathssolutions" on the 9th of August 2021.
A semicircle is drawn at one side of a quarter circle internally. A circle touching semicircle, another side of quarter circle, and curve of a quarter circle. Let angle between the line joining the center of small circle and semicircle and side of the quarter circle is 'y'. Then cos(y) = ?
Take a look at the figure given below. Give it a try and when you are ready then watch the solution.
Step 1: (Constructions)
Let Radius of quarter circle be R.
Radius of semicircle be R/2.
Radius of small circle be r.
Join center of small circle and quarter circle.
Draw perpendicular from center of small circle to both sides of quarter circle
Angle forming by line joining center of small circle and quarter circle be x
Length of line joining center of small circle and quarter circle is R - r
Length of line joining center of small circle and semicircle is R/2 + r
Perpendicular from center of small circle to Bottom side of quarter circle is r
Step 3: (Calculations)
At right angled triangle with one angle x and two sides r, (R - r):
\[cos(x)=\frac{r}{R-r}\]
At angle x in triangle with sides R/2, (R - r), (R/2 + r)
\[cos(x)=\frac{\left(\frac{R}{2}\right)^2+(R-r)^2-\left(\frac{R}{2}+r\right)^2}{2\left(\frac{R}{2}\right)(R-r)}\]
By comparing both equations,
\[\Rightarrow\frac{r}{R-r}=\frac{\left(\frac{R}{2}\right)^2+(R-r)^2-\left(\frac{R}{2}+r\right)^2}{2\left(\frac{R}{2}\right)(R-r)}\]
\[\Rightarrow \frac{r}{1}=\frac{\left(\frac{R}{2}\right)^2+(R-r)^2-\left(\frac{R}{2}+r\right)^2}{R}\]
\[\Rightarrow Rr=\frac{R^2}{4}+R^2+r^2-2Rr-\frac{R^2}{4}-r^2-Rr\]
After some cancels outs,
\[\Rightarrow Rr=R^2-3Rr\]
\[\Rightarrow 4Rr=R^2\]
\[\Rightarrow r=\frac{R}{4}\]
Step 4:
Observe triangle with angle y and sides (R/2 - r), (R/2 + r)
\[\Rightarrow cos(y)=\frac{\frac{R}{2}-r}{\frac{R}{2}+r}\]
\[\Rightarrow cos(y)=\frac{\frac{R}{2}-\frac{R}{4}}{\frac{R}{2}+\frac{R}{4}}\]
\[\Rightarrow cos(y)=\frac{\frac{R}{4}}{\frac{3R}{4}}\]
\[\Rightarrow cos(y)=\frac{1}{3}\]
By this, I got cos(y) = 1/3
Here is Image solution and calculations for this puzzle
