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

$2 r \sin α = 22$

$2 r \sin 2 α - 2 r \sin 3 α = 20$

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

$2 r \sin α \cos α - r \sin α (3 - 4 \sin^{2} α) = 10$

i.e. $22 \cos α - 11 (4 \cos^{2} α - 1) = 10$

i.e. $1 + 22 \cos α - 44 \cos^{2} α = 0$

and now solve for $\cos α$ . 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 $\cos α$ , do you just go find the values of everything else (radius, $\cos 2 α$ 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

2 r \sin 3 α

and recall that

r = 11 / \sin α

. You need not find the sines of the angles, since

\sin 3 α = \sin α (4 \cos^{2} α - 1)

Kerwin