geometry question

• Jun 30th 2012, 12:47 PM
Pat722
geometry question
Given unit circle P. Points A and B are on the circle (r=1), forming an acute angle APB (theta). What line segment is equal to the tangent theta if line segment DC is tanget to circle at point b.
• Jun 30th 2012, 03:08 PM
Soroban
Re: geometry question
Hello, Pat722!

The directions are somewhat fuzzy.
I may have misinterpreted them.

Quote:

$\displaystyle \text{Given unit circle }P\text{, points }A,B\text{ are on the circle, forming}$
$\displaystyle \text{acute angle }AP\!B \,=\,\theta.\;\text{What line segment is equal to }\tan\theta$
$\displaystyle \text{if line segment }DC\text{ is tangent to circle at point }B\,?$

I believe the diagram looks like this:
Code:

                          E                           o                         .:                         . :                       .  :               * * *  A.  :           *          o    o D         *          /  * |       *        1 /    *|                   /      |       *          / @      *       *        o - - - - o B       *      P      1    *                           |       *                *|         *              * |           *          *  o               * * *      C
We have the unit circle with center $\displaystyle P$ and radii $\displaystyle P\!A = PB = 1.$
$\displaystyle \theta \,=\,\angle AP\!B,\;\theta \,<\,90^o$
$\displaystyle CD$ is tangent to the circle at $\displaystyle B.$
Extend $\displaystyle CD$ to meet $\displaystyle P\!A$ extended at $\displaystyle E.$

Then: .$\displaystyle \tan\theta \,=\,\frac{EB}{1}$

Therefore: .$\displaystyle EB \:=\:\tan\theta$