# Thread: Angle bisection problem, with very few measures

1. ## Angle bisection problem, with very few measures

Hi Forum!

I've been introduced to the Law of Tangents and I have some doubts about it.

In $\Delta ABC$, let D be a point in BC such that AD bisects angle A. Given that AD=6, BD=4, and DC=3, find AB.

Is there any other possibility?
Is the Law of Tangents really needed here? If so, how do we use it?

Thanks!

2. Hi Zellator,
Did the problem state AD =6 or AC=6 ?

3. Originally Posted by bjhopper
Hi Zellator,
Did the problem state AD =6 or AC=6 ?
Hi bjhopper!
No, that's all!
The question is here Law of Tangents - AoPSWiki
It was a question of Mu Alpha Theta. They have really great questions.
Unfortunately, I can't find the solved test from 1991.

It involves the Law of Tangents, nevertheless.

4. Code:
   A

C      D        B
Given: AD = 6, CD = 3, BD = 4 ; AD bisects angleBAC

AC = 6, AB = 8

5. Hi Wilmer!
What method did you used?
How did you know that triangle ACD was isosceles?
Did you found this by graphing?

6. Hi Zellator,
Wilmer shows the use of the angle bisector theorem which is what I thought the problem was supposed to be.I looked at your reference which shows it as unedited.The problem is trig not geometry,My own opinion is that the law of tangents is another way to solve a triangle given two sides and included angle (rather than cosine law)

7. Doesn't the angle bisector theorem works only for right triangles?
Thank you both for your help.

8. Bisector theorem applies to all triangles. AB / BD = AC/DC. Can you prove it?

9. Originally Posted by Zellator
Doesn't the angle bisector theorem works only for right triangles?
Well, NEVER "assume"; draw a non-right triangle and TRY it.

10. I had to edit the post several times, because it was actually wrong.
The right is CB/AC=AC/CD.
Proof...
If we consider that the angles ADB and angle ADC are supplementary; their sines will be equal.
And since we have AB/BD=sin angle ADB/sin angle DAB and AC/DC=sin angle ADC/sin angle DAC.
So AB/BD=AC/DC
Or something along this lines.
Thanks bjhopper and Wilmer!
I got it now, it was a minor mistake here.

Originally Posted by Wilmer
Well, NEVER "assume"; draw a non-right triangle and TRY it.
I'm sorry Wilmer.
That is not how I usually think about a problem. I need to have some base in knowledge before I can TRY anything.
Therefore, I never ASSUME something is right without proof.

11. OKkkkkk....if you're "collecting" interesting ones:

triangle ABC with sides BC = 24, AC = 40 and AB = 56

D on BC, E on AC, F on AB such that AD, BE and CF are angle bisectors

BC: BD = 14, CD = 10
AC: AE = 28, CE = 12
AB: AF = 35, BF = 21

In other words, ALL integers.

Have fun(?): try case where triangle ABC = 70-105-140.

12. Google. Proof of angle bisector theorem and then to hotmath.com.

13. Hi bjhopper!
All examples I saw where with right triangles, that confused me. But now I got it.
Thanks for the site recommendation, this is great!
I learn much faster with solved problems.

14. ## Re: Angle bisection problem, with very few measures

Hey Guys!
Sorry, I had to resuscitate this thread. Since I still don't get it, really sorry about that.

The question is, how does $AD$ comes into play?
We have $\frac {AB}{BD}=\frac{AC}{DC}$

I can only find the bisector theorem in this terms...

I don't know what the the length of the angle bisector equals so it I really don't know how to use it.
It is present in Stewart's Theorem, but to use it we would need another leg.
Can someone do it step by step? That would be really appreciated. Thanks.

15. ## Re: Angle bisection problem, with very few measures

Hi Zellator,
If I understand your new post you want to know how to find the lengths of AB and AC using Stewards Theorem. I am not familar with it but I have a plan for you.
Draw a vertical line length 6 Mark upper end A and lower D.At point D swing two arcs lengths of 3 and 4.A straight line thru D defines two points B and C.Note that you can draw many triangles.If you fix the length of either AB or AC the other line can be calculated using the cosine law

bjh

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