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        riddles >> easy >> An angle in a triangle
        
(Message started by: jollytall on Jan 10^th, 2013, 10:58am)

Title: An angle in a triangle
Post by jollytall on Jan 10^th, 2013, 10:58am

Given the ABC triangle, where AB=BC and ABC=20°.
On AB we choose D, that DCB=20° and on BC we choose E, that EAB=30°. What is BDE angle?

This was given on a 7th grade (13 years) math competition. With a lot of math I could calculate the solution up to lots of digits, still could not prove the intended "round" number.
(No child could solve it either.)

Title: Re: An angle in a triangle
Post by towr on Jan 10^th, 2013, 1:39pm

Well, cinderella (http://www.cinderella.de/tiki-index.php?page=Download+Cinderella.2) can find it easily enough. So if a dumb computer can do it, there must be a way.

Title: Re: An angle in a triangle
Post by jollytall on Jan 11^th, 2013, 3:58am

Yes, as it is a definite structure, it has a definite answer.
Even I could calculate it, but got an arcsin(...) complicated formula, that gave the intended solution with a high precisity. Nonetheless I do not consider it a solution, since the intended answer is a "nice", "round" number.
Having asked it on a 7th grade competition, I would expect an easy and nice solution.

Title: Re: An angle in a triangle
Post by rmsgrey on Jan 11^th, 2013, 4:28am

My guess would be that there's either a similar problem with an easy solution, or an almost correct easy solution - and whoever set the question made a mistake.

On the other hand, depending on how hard the competition was meant to be, getting a nice solution by a nasty route is not inconceivable.

Title: Re: An angle in a triangle
Post by Grimbal on Jan 11^th, 2013, 5:47am

If it is a nice round number, you can just make a precise picture and measure the angle.

Title: Re: An angle in a triangle
Post by towr on Jan 11^th, 2013, 6:06am

See attached drawing, line k and d are parallel by construction (via obvious duplication, rotation etc of starting triangle).

Most is clear enough from the picture, but we have to prove that line k intersects at D (i.e. D' = D)

We have from the sine rule
CD/sin(80) = AC/sin(40)
BD'/sin(50) = BF/sin(30)

From the double angle formula, we have
sin(80) = 2 * cos(40) * sin(40)
=>
sin(80) = 2 * sin(50) * sin(40)
=>
sin(80) = 1/sin(30) * sin(50) * sin(40)

therefor
BD' = CD = BD, so line k does indeed cross at D

So finally BDE = FDC = 180-30-40 = 110

So, admittedly, it took me three different approaches, and probably a few hours, but it doesn't require any arcane knowledge.

Title: Re: An angle in a triangle
Post by jollytall on Jan 11^th, 2013, 8:44am

Thanks, very nice solution, (though not for a 7th grader).
I got to a similar point, but could not prove that D=D'.
The other way that I tried was that if f and c crosses at G, then DC=CG. That would also give away the solution, but again, I could not prove.

Title: Re: An angle in a triangle
Post by jollytall on Jan 12^th, 2013, 10:43am

Finally I got (from another father) a really elegent solution (see attached).

Let's mirror ABC to AB and AC respectively.

AC'=AB' and C'AB'=60° therefore C'B'=AB (green) too.

ACM is the 30° line and therefore AC'M and its extention AC'P as well (blue lines) . Also PC'B' must be 30°, i.e. C'P is the angle halving line with AP=PB' and APC'=90°. O is the crossing point of C'P and AC. Because PO is the AB' side halving right angle line, AO=OB'.

On the other hand NBA=NB'A=B'AN=20° (red lines). Consequently AN=NB'.

It is only possible if N=O (I intentionally drew it incorrectly). So the questioned MN section is the same as the MO section. From there POA=70°, AOM=ANM=110°and NMA=50°.

Title: Re: An angle in a triangle
Post by towr on Jan 12^th, 2013, 11:47am

Nicely done.

Title: Re: An angle in a triangle
Post by Mariko79 on Apr 8^th, 2013, 11:25pm

on 01/10/13 at 10:58:10, jollytall wrote:

This was given on a 7th grade (13 years) math competition. With a lot of math I could calculate the solution up to lots of digits, still could not prove the intended "round" number.
(No child could solve it either.)

It's normal...this is too complex for the 13 years childrens... . (http://www.directory.bodybuildingtips-list.com/Business.html)

Title: Re: An angle in a triangle
Post by Rosiethomas on Jul 1^st, 2013, 3:29am

110

I agree this was a difficult one for 13 year old kid. You can use these formulae

transtutors.com/math-homework-help/laws-of-triangle/properties-of-triangle.aspx