# Thread: Two secants to a circle

1. ## Two secants to a circle

BA^2 + CD^2 = 16
PA = 1.25
PC = 0.75
AB? CD?

Thank you!

2. ## Re: Two secants to a circle

What have you tried?
Do you know the 'intersecting secant theorem'?
Look here:
Intersecting Secant Theorem - Math Open Reference

3. ## Re: Two secants to a circle

Originally Posted by Siron
What have you tried?
Do you know the 'intersecting secant theorem'?
Look here:
Intersecting Secant Theorem - Math Open Reference
However I can't resolve the problem anyway, in fact I only know something about AB and CD and not about PB and PD. What do you think?

4. ## Re: Two secants to a circle

Along the theorem:
$PA\cdot PB = PC \cdot PD$
But we can write $PB=PA+AB$ and $PD=PC+DC$ therefore:
$PA\cdot (PA+AB)=PC \cdot (PC+DC)$
$\Leftrightarrow (PA)^2+PA\cdot AB=(PC)^2+PC\cdot DC$
$\Leftrightarrow DC=\frac{(PA)^2+PA\cdot AB - (PC)^2}{PC}$ (1)

You have given $PA=1,25$ and $PC=0,75$ so you can substitute them in (1). You have also given:
$(AB)^2+(DC)^2=16$ (2)

If we susbtitute (1) in (2) we get:
$(AB)^2+\left[\frac{(PA)^2+PA\cdot AB - (PC)^2}{PC}\right]^2=16$

If you have substituted the given values for PC and PA into this equation then you will have a quadratic equation in one variable AB which you can solve (you will get two solutions but offcourse you have to reject the negative one because the lenght of a side can't be negative).

5. ## Re: Two secants to a circle

Wow, very nice explanation, thank you very much!
Last thing: I looked for a demonstration of the intersecting secant theorem but I couldn't find one, how can I get to that theorem?

6. ## Re: Two secants to a circle

Originally Posted by goby
Wow, very nice explanation, thank you very much!
Last thing: I looked for a demonstration of the intersecting secant theorem but I couldn't find one, how can I get to that theorem?
You're welcome! Did you find AB and CD?
If you want a proof then you can take a look here:
http://www.mathforamerica.org/c/docu...e=DLFE-112.pdf

7. ## Re: Two secants to a circle

Sure! I solved the equation and I get AB = 1,44 so CD is 3.73. Thank you again!