Question

if pa=10 and be=21 find bp

Answer 1

Is there a diagram to go with this information?

Answer 2

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

Answer 3

Yes

Answer 4

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.

Answer 5

It's not very direct, but will work.

Answer 6

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

Answer 7

Angle AOP being acute.

Answer 8

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

Answer 9

Oh yeah...

Answer 10

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

Answer 11

OA=10.5 correct?

Answer 12

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.

Answer 13

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

Answer 14

I'm still not sure how to find EP?

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

(take square root of c^2)

Answer 19

Ahhh. 14.5 + 10.5 correct?
Equaling 25

Answer 20

If you use the theorem from that website I linked to, it shows the following:

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.

Answer 21

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.

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

Sure, I got time.

Answer 26

Yes, and the PT is very easy.

Answer 27

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

Answer 28

Exactly.

Answer 29

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.

Answer 30

Will do.