Dallasb22 3 years ago if pa=10 and be=21 find bp

1. CliffSedge

Is there a diagram to go with this information?

2. Dallasb22

3. CliffSedge

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

4. Dallasb22

Yes

5. CliffSedge

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.

6. CliffSedge

It's not very direct, but will work.

7. Dallasb22

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

8. Dallasb22

Angle AOP being acute.

9. CliffSedge

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

10. Dallasb22

Oh yeah...

11. CliffSedge

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

12. Dallasb22

OA=10.5 correct?

13. CliffSedge

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.

14. CliffSedge

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

15. Dallasb22

|dw:1346780775442:dw|

16. Dallasb22

I'm still not sure how to find EP?

17. CliffSedge

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

18. Dallasb22

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

19. Dallasb22

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

20. CliffSedge

(take square root of c^2)

21. Dallasb22

Ahhh. 14.5 + 10.5 correct? Equaling 25

22. CliffSedge

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.

23. CliffSedge

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.

24. Dallasb22

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

25. Dallasb22

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

26. CliffSedge

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

27. CliffSedge

Sure, I got time.

28. Dallasb22

Yes, and the PT is very easy.

29. CliffSedge

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

30. Dallasb22

Exactly.

31. CliffSedge

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.

32. Dallasb22

Will do.