• Dec 11th 2009, 04:58 AM
reiward
Each of two tangents from an external point to a circle is 3cm long. The smaller arc which they intercept is 112degrees 30' 26''. Find the radius of the circle.

Help please, I really have no idea. Can you show it with illustrations, thanks!
• Dec 11th 2009, 05:37 AM
Soroban
Hello, reiward!

Quote:

Each of two tangents from an external point to a circle is 3 cm long.
The smaller arc which they intercept is 112° 30' 26''.
Find the radius of the circle.

Code:

              * * *           *          *  A         *              o       *          r  *  * *    3                     *        *       *          * θ    *      *       *      C o - - - - * - - - - o P       *          *      *      *                     *        *       *          r  *  * *    3         *              o           *          *  B                 * * *

The center of the circle is $\displaystyle C$; its radius is $\displaystyle r.$

Tangents $\displaystyle PA = PB = 3$
$\displaystyle \angle ACB \,=\,112^o30'13'' \quad\Rightarrow\quad \theta \:=\:\angle ACP \:\approx\:56.2536^o$

In right triangle $\displaystyle CAP\!:\;\;\sin\theta \:=\:\frac{3}{r} \quad\Rightarrow\quad r \:=\:\frac{3}{\sin\theta}$

ThereforeL .$\displaystyle r \:=\:\frac{3}{\sin56.2536^o} \;\approx\;3.6\text{ cm}$