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        riddles >> medium >> Geometry: Find angle in 50-60-70 triangle
        
(Message started by: Aryabhatta on Dec 5th, 2007, 10:47pm)
Title: Geometry: Find angle in 50-60-70 triangle
Post by Aryabhatta on Dec 5th, 2007, 10:47pm
In the attached figure, ABC is a triangle such that <A = 70, <B = 60 and <C = 50.

D is chosen on AC such that <ABD = 45 and E is chosen on BC such that <BDE = 50.

Find <AED.
Title: Re: Geometry: Find angle in 50-60-70 triangle
Post by ThudanBlunder on Dec 5th, 2007, 11:01pm
Looks familiar (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_medium;action=display;num=1069922798).

See also http://mathcircle.berkeley.edu/BMC4/Handouts/geoprob.pdf
Title: Re: Geometry: Find angle in 50-60-70 triangle
Post by Aryabhatta on Dec 5th, 2007, 11:42pm
Err.. Yes!

This problem was inspired by the problem in the thread you mention, which I received from a different source. (Rather it was inspired by the proof I came up with for the other problem).

Doing some research based on the link for the pdf you gave it seems like Alexander Kornienko had a proof very similar to what I have (for the 80-20-80 version).

These kind of problems seem to have been beaten to death it seems. Sorry about that  :-[

If anyone wants to, I can edit this post later to attach the image of the proof I had, though the link by T&B and the name above should be enough to come up with a proof...
Title: Re: Geometry: Find angle in 50-60-70 triangle
Post by ThudanBlunder on Dec 6th, 2007, 12:17am

on 12/05/07 at 23:42:17, Aryabhatta wrote:
Sorry about that  :-[/quote]
No, I think it's a really nice problem and worth disinterring. When I was in China teaching English I gave it to several classes, from schoolkids to engineering Masters students. In order to motivate them I offered RMB50 (about US$6, but worth more in China) to the first person who could solve it before the end of class, ie. within about 20 minutes. It was funny watching as their confidence in solving an apparently easy geometry problem turned to irritated befuddlement and frustration. Only one student, a schoolboy at that, managed to win my RMB50. And not only did he solve it, but he also did it without trig.

[quote author=Aryabhatta link=board=riddles_medium;num=1196923638;start=0#2 date=12/05/07 at 23:42:17]These kind of problems seem to have been beaten to death it seems.

Well, one can still have a go at the general solution.    ;)
Title: Re: Geometry: Find angle in 50-60-70 triangle
Post by tiber13 on Dec 7th, 2007, 9:59am
draw it out, then measure the angles.
Title: Re: Geometry: Find angle in 50-60-70 triangle
Post by Aryabhatta on Dec 7th, 2007, 1:09pm

on 12/07/07 at 09:59:40, tiber13 wrote:
draw it out, then measure the angles.


So, what did you get?
Title: Re: Geometry: Find angle in 50-60-70 triangle
Post by Hippo on Dec 8th, 2007, 2:01am

on 12/06/07 at 00:17:14, ThudanBlunder wrote:
Well, one can still have a go at the general solution.    ;)


I suppose there is nothing special with general solution: Use sine theorem to compute unknown in e(edge)a(angle)e determined triangles and use cosine theorem to compute unknown in aea determined triangles (choose one distance arbitrary). The problem I expected is the expressions will become rather long ...

Method to draw the triangle and measure the angle would be the fastest ... especialy if you do it with METAFONT/METAPOST like program ;)
Title: Re: Geometry: Find angle in 50-60-70 triangle
Post by cool_jansen on Dec 13th, 2007, 9:55am
Here's my solution

Let AB be 10 units

Look at triangle ADB

<ADB= 65
using sine rule, sin <ADB/10 = sin 70/BD ,
   ==> BD = 10*sin70/sin65 = 8.5165 = X

                       
    AD/sin45 = 10/sin 65 ==> AD=7.80206=Y


Look at triangle BDE
<DEB= 115
BD/sin 115 = DE/sin15
==> DE =X*sin 15/sin115 =2.4321 = Z

Now, look at triangle ADE
<ADE=115
using cosine rule,
AE=(Y^2 + Z^2 - 2*Y*Z*cos115)^0.5=9.10088=W

So, sin <AED/Y=sin115/W
==>sin<AED=Y sin 115/W = 0.776965

so, <AED = 50.99= 51.0

1st time posting, I had difficulties writing various math symbols here... haha ;D[hideb][/hideb]
Title: Re: Geometry: Find angle in 50-60-70 triangle
Post by Aryabhatta on Dec 13th, 2007, 10:57am
You are close cool_jansen.

I think the answer is exactly 50.
Title: Re: Geometry: Find angle in 50-60-70 triangle
Post by cool_jansen on Dec 13th, 2007, 5:00pm

on 12/13/07 at 10:57:59, Aryabhatta wrote:
You are close cool_jansen.

I think the answer is exactly 50.


Interesting. Any hint for the approach used for this question? And did u spot any fault in my solution?
Title: Re: Geometry: Find angle in 50-60-70 triangle
Post by Aryabhatta on Dec 14th, 2007, 2:14am

on 12/13/07 at 17:00:26, cool_jansen wrote:
Interesting. Any hint for the approach used for this question? And did u spot any fault in my solution?


For a hint, look at the pdf link which ThudanBlunder gave and look for the name "Alexander Kornienko" which I mentioned in a previous post.

I did not consciously try to look for a flaw in your solution, as I know that the answer is 50.
Title: Re: Geometry: Find angle in 50-60-70 triangle
Post by JiNbOtAk on Dec 15th, 2007, 9:48pm

on 12/06/07 at 00:17:14, ThudanBlunder wrote:
Only one student, a schoolboy at that, managed to win my RMB50. And not only did he solve it, but he also did it without trig.


Really ? How did he solved it then ?

Note : Personally, I like tiber13's answer. Imaginative, reminded me how edison measured the volume of a lightbulb he invented.
Title: Re: Geometry: Find angle in 50-60-70 triangle
Post by Aryabhatta on Dec 16th, 2007, 2:29am

on 12/15/07 at 21:48:38, JiNbOtAk wrote:
Really ? How did he solved it then ?



Did you check out the pdf link given by T&B? There are many non-trig solutions for the 80-20-80 version. In fact the solution I have does not use trigonometry at all. The same applies to the problem of the current thread.
Title: Re: Geometry: Find angle in 50-60-70 triangle
Post by ThudanBlunder on Dec 17th, 2007, 1:02am

on 12/15/07 at 21:48:38, JiNbOtAk wrote:
Really ? How did he solved it then ?

Like this (http://agutie.homestead.com/files/LangleyProblem.html).
Title: Re: Geometry: Find angle in 50-60-70 triangle
Post by JiNbOtAk on Dec 17th, 2007, 1:23am

on 12/17/07 at 01:02:00, ThudanBlunder wrote:
Like this (http://agutie.homestead.com/files/LangleyProblem.html).


Nice, very nice. And your student came up with that working step ?  Impressive..


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