An arc which subtends and angle of 60° at O is shown below. Find the ratio of r : R.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyK6BdrrDmtv8qzfFGYXxMkDQKOG5MB9Zd1Tsvm3Gu_Goth6RBphXmRv2ovYX-ndwnWLmhfc1YgggMNRWLrpOLqZx-pVOZs9gjFsB9j4rFjfAjs4GqZgp17KG7MpsJnr7L6jcDD8KEXsrl/s400/Circle+and+trigonometry.jpg)
Select the correct answer from the following choices.
i) 1 : 2
ii) 1 : √2
iii) 1 : √3
iv) 1 : 3
Solutions:
With reference to the following diagram;
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhyphenhyphenbePfoIVgL3_PIysAh8fAvfh8S2zzvqhkC7IVoviovZP9rDyIYUPE511puyoTI58L-t-maNinpkstB1A4X7d2U-nf2P09Jtrhe9VIOMZU7I5R0ry043Fs_xTOVi3UsfGzAkwtMBmzgyR/s400/Circle+and+trigonometry+-+1.jpg)
The radius ABO passed through the center of the circle ARC and bisects the angle EOD.
Triangle BCO is a right angled triangle with angles 30°, 60° and 90°.
By trigonometry,
if BC = r, then BO = 2r.
Therefore radius of the arc DAE = AB + BO = 3r
Given that the radius of the arc DAE = R,
That is, R = 3r
Therefore the ratio of r : R is 1 : 3.
The answer is (iv).
Reference
- Hey Math! assignment (E-Maths), July 2010, Question #7, Circle and trigonometry