if pa=10 and be=21 find bp

- anonymous

if pa=10 and be=21 find bp

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- anonymous

Is there a diagram to go with this information?

- anonymous

##### 1 Attachment

- anonymous

Thank you. Can we assume AP is tangent to the circle and that BE is a diameter?

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## More answers

- anonymous

Yes

- anonymous

Well, first thing that comes to mind (because I can't remember the tangent-chord theorems off the top of my head right now..) is that you can form a right triangle AOP with O the center of the circle and AO a radius. From there you can use Pythagoras' theorem to get the missing lengths.

- anonymous

It's not very direct, but will work.

- anonymous

How can AOP form a right triangle?
I drew the line(auxillary lines) and it appears acute.

- anonymous

Angle AOP being acute.

- anonymous

Yes. Angle OAP is the right angle because tangents to a circle are always at right angles to their radii.

- anonymous

Oh yeah...

- anonymous

I think that's the preliminary proposition that is used to prove the various tangent and chord theorems. I like this website for reviewing such things:
http://www.mathwarehouse.com/geometry/circle/tangent-secant-side-length.php

- anonymous

OA=10.5 correct?

- anonymous

Line OEP is therefore the hypotenuse, and when you add OB=OA, you get BP.
Yes, OA is a radius so is half the diameter BE.

- anonymous

Oh, that's nice, everything works out to rational numbers. Very kind of them . . .

- anonymous

|dw:1346780775442:dw|

- anonymous

I'm still not sure how to find EP?

- anonymous

I don't think you need EP, but did you use Pythagoras' theorem to get OP?

- anonymous

10(^2)+10.5(^2)=c(^2)
I get 210.25???

- anonymous

I was going to find EP. Then add BE+EP=BP

- anonymous

(take square root of c^2)

- anonymous

Ahhh. 14.5 + 10.5 correct?
Equaling 25

- anonymous

If you use the theorem from that website I linked to, it shows the following:
|dw:1346781296754:dw|
Which when you simplify and put into standard form yields the quadratic equation
x^2+21x-100=0.
I find that unnecessary if you don't like quadratics. Either way it includes an extra step and Pythagoras' theorem gets us there just as easily.

- anonymous

Yes, the solutions to that quadratic equation are x=4 and x=-25. Throw out the negative length as meaningless, and EP=4; add that to the diameter and BP=25.

- anonymous

Okay good.
Sometimes I just need someone to make me ask myself the right questions.

- anonymous

Thanks for helping! I have another if you don't mind...?

- anonymous

I usually never bother to remember such specific theorems, I prefer to deduce everything from more basic stuff that is easy to remember.

- anonymous

Sure, I got time.

- anonymous

Yes, and the PT is very easy.

- anonymous

Exactly, it shows up everywhere; only thing you had to remember is that tangents are perpendicular to radii.

- anonymous

Exactly.

- anonymous

Might as well start a new thread for your next question; maybe we can get others to jump in and add their insights as well.

- anonymous

Will do.

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