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Topic: Geometry: Find angle in 50-60-70 triangle (Read 4816 times) |
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Aryabhatta
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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.
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ThudnBlunder
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Re: Geometry: Find angle in 50-60-70 triangle
« Reply #1 on: Dec 5th, 2007, 11:01pm » |
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Looks familiar.
See also http://mathcircle.berkeley.edu/BMC4/Handouts/geoprob.pdf
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| « Last Edit: Dec 5th, 2007, 11:21pm by ThudnBlunder » |
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Aryabhatta
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Re: Geometry: Find angle in 50-60-70 triangle
« Reply #2 on: Dec 5th, 2007, 11:42pm » |
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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...
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| « Last Edit: Dec 5th, 2007, 11:45pm by Aryabhatta » |
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ThudnBlunder
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Re: Geometry: Find angle in 50-60-70 triangle
« Reply #3 on: Dec 6th, 2007, 12:17am » |
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on Dec 5th, 2007, 11:42pm, Aryabhatta wrote:| Sorry about that |
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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.
on Dec 5th, 2007, 11:42pm, Aryabhatta wrote:| These kind of problems seem to have been beaten to death it seems. |
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Well, one can still have a go at the general solution. 
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| « Last Edit: Dec 8th, 2007, 4:00pm by ThudnBlunder » |
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Random Lack of Squiggily Lines
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Re: Geometry: Find angle in 50-60-70 triangle
« Reply #4 on: Dec 7th, 2007, 9:59am » |
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draw it out, then measure the angles.
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You can only believe i what you can prove, and since you have nothing proven to cmpare to, you can believe in nothing.
I have ~50 posts to hack a "R" into a "D". Which one?
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Aryabhatta
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Re: Geometry: Find angle in 50-60-70 triangle
« Reply #5 on: Dec 7th, 2007, 1:09pm » |
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on Dec 7th, 2007, 9:59am, tiber13 wrote:draw it out, then measure the angles.
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So, what did you get?
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Hippo
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Re: Geometry: Find angle in 50-60-70 triangle
« Reply #6 on: Dec 8th, 2007, 2:01am » |
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on Dec 6th, 2007, 12:17am, ThudanBlunder wrote:
Well, one can still have a go at the general solution.
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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
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| « Last Edit: Dec 8th, 2007, 2:02am by Hippo » |
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cool_jansen
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Re: Geometry: Find angle in 50-60-70 triangle
« Reply #7 on: Dec 13th, 2007, 9:55am » |
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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  [hideb][/hideb]
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Aryabhatta
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Re: Geometry: Find angle in 50-60-70 triangle
« Reply #8 on: Dec 13th, 2007, 10:57am » |
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You are close cool_jansen.
I think the answer is exactly 50.
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cool_jansen
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Re: Geometry: Find angle in 50-60-70 triangle
« Reply #9 on: Dec 13th, 2007, 5:00pm » |
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on Dec 13th, 2007, 10:57am, Aryabhatta wrote:You are close cool_jansen.
I think the answer is exactly 50. |
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Interesting. Any hint for the approach used for this question? And did u spot any fault in my solution?
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Aryabhatta
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Re: Geometry: Find angle in 50-60-70 triangle
« Reply #10 on: Dec 14th, 2007, 2:14am » |
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on Dec 13th, 2007, 5:00pm, cool_jansen wrote:
Interesting. Any hint for the approach used for this question? And did u spot any fault in my solution? |
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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.
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JiNbOtAk
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Re: Geometry: Find angle in 50-60-70 triangle
« Reply #11 on: Dec 15th, 2007, 9:48pm » |
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on Dec 6th, 2007, 12:17am, 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. |
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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.
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Aryabhatta
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Re: Geometry: Find angle in 50-60-70 triangle
« Reply #12 on: Dec 16th, 2007, 2:29am » |
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on Dec 15th, 2007, 9:48pm, JiNbOtAk wrote:
Really ? How did he solved it then ?
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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.
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ThudnBlunder
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Re: Geometry: Find angle in 50-60-70 triangle
« Reply #13 on: Dec 17th, 2007, 1:02am » |
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on Dec 15th, 2007, 9:48pm, JiNbOtAk wrote:
Really ? How did he solved it then ?
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Like this.
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JiNbOtAk
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Re: Geometry: Find angle in 50-60-70 triangle
« Reply #14 on: Dec 17th, 2007, 1:23am » |
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on Dec 17th, 2007, 1:02am, ThudanBlunder wrote:
Nice, very nice. And your student came up with that working step ? Impressive..
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