# Thread: geometry question

1. ## 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.

2. ## Re: geometry question

Hello, Pat722!

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

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

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 $P$ and radii $P\!A = PB = 1.$
$\theta \,=\,\angle AP\!B,\;\theta \,<\,90^o$
$CD$ is tangent to the circle at $B.$
Extend $CD$ to meet $P\!A$ extended at $E.$

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

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