Welcome to NRICH.

 
Length of Chord in Circle
By Anthony Cardell Tony on Wednesday, January 02, 2002 - 03:25 am:

I was wondering if anyone could help me with the solution to this problem:

An a-degree arc makes a chord of 22 cm (radius of circle upon which chord is constructed is not given). A 3a degree arc makes a chord 20 centimeters shorter than a 2a degree arc. What is the length of the 3a degree chord?

Thanks in advance

By Kerwin Hui on Wednesday, January 02, 2002 - 05:55 pm:

Let r be the radius of the circle.

We have the conditions
2r sin a=22
2r sin 2a-2r sin 3a=20

where a=a/2. Now the second equation can be written as

2r sin a cos a-r sin a(3-4 sin2 a) =10

i.e. 22 cos a-11(4 cos2a-1)=10
i.e. 1+22 cos a-44 cos2a=0

and now solve for cosa. From here, it is easy to find the length of our chord.

Kerwin

By Anthony Cardell Tony on Thursday, January 03, 2002 - 03:13 am:

Thanks for the help!

I just have one question about the solution.
Once you have cosa, do you just go find the values of everything else (radius, cos2a etc.) based on this, and then plug that into the final answer, or is there a shortcut that I don't see?

By Kerwin Hui on Thursday, January 03, 2002 - 07:17 am:

The chord has length 2rsin3a and recall that r=11/sina. You need not find the sines of the angles, since sin 3a=sin a (4 cos2 a-1)

Kerwin