O is the center of a circle. AB, CD and EF are three parallel chords of this circle having respective lengths 2, 3 and 4.
It is known that ∠AOB = m, ∠COD = n and, ∠EOF = m+n, where m+n < 180o
Determine cos m
What is really needed is the radius of the circle since by the law of cosines cos(m)=(2r²-4)/(2r²)
(I first used the law of cosines for cos(n) and cos(n+m) and then the cosine of a sum formula gives and equation. Solving this by graphing calculator I found r². But the following is a better way.)
If you rotate the chords the central angles don't change. So go ahead and rotate them until a triangle of sides 2, 3, 4 is formed. Circle O is then the circumcircle for this triangle. Here's a nifty formula for the radius of the circumcircle: r = abc/(4*area)
By Heron's formula the area = √(135/16)
which yields
r²=64/15
and
cos(m)=17/32
(Which is also what I found doing it the hard way.)
|
|
Posted by Jer
on 2015-12-06 12:55:48 |
Two solutions
