This question was uploaded on 10/10/21 on social media accounts.
A circle is inscribed in a semicircle and also separated by a chord perpendicular to the diameter of the semicircle. Half of the length of the chord is 12 and the diameter is 25. Find the area of a circle.
Solution:
This is a standard result, click here to get its proof.
By the standard result:
\[r = 2R\cdot\sinθ\cdot(1 - \sinθ)\]
By intersecting chord theorem:
\[x(25-x)=12^2\]
\[\Rightarrow x=9⠀or⠀16\]
Clearly, x is less than the radius,
\[\Rightarrow x=9\]
\[\Rightarrow 25-x=16\]
\[\Rightarrow \tan\theta=\frac{12}{16}=\frac{3}{4}\]
\[\Rightarrow \sin\theta=\frac{3}{5}\]
\[\Rightarrow r = 25\cdot\frac35\cdot(1 - \frac35)\]
\[\Rightarrow r = 6\]
\[\Rightarrow Area = 36\pi\]
If we consider x can be greater than the radius then,
\[\sin\theta = \frac{4}{5}\]
\[r = 4\]
\[Area = 16\pi\]
Puzzle related to Geometry, Area, Circle.
