Author |
Topic: An angle in a triangle (Read 8928 times) |
|
jollytall
Senior Riddler
Gender: 
Posts: 585
|
 |
An angle in a triangle
« on: Jan 10th, 2013, 10:58am » |
 Quote  Modify
|
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.)
|
|
 IP Logged |
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: An angle in a triangle
« Reply #1 on: Jan 10th, 2013, 1:39pm » |
Quote Modify
|
Well, cinderella can find it easily enough. So if a dumb computer can do it, there must be a way.
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
jollytall
Senior Riddler
Gender:
Posts: 585
|
|
Re: An angle in a triangle
« Reply #2 on: Jan 11th, 2013, 3:58am » |
Quote Modify
|
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.
|
|
IP Logged |
|
|
|
rmsgrey
Uberpuzzler
    
Gender:
Posts: 2873
|
|
Re: An angle in a triangle
« Reply #3 on: Jan 11th, 2013, 4:28am » |
Quote Modify
|
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.
|
|
IP Logged |
|
|
|
Grimbal
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 7527
|
|
Re: An angle in a triangle
« Reply #4 on: Jan 11th, 2013, 5:47am » |
Quote Modify
|
If it is a nice round number, you can just make a precise picture and measure the angle.
|
|
IP Logged |
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
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.
|
| « Last Edit: Jan 11th, 2013, 6:17am by towr » |
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
jollytall
Senior Riddler
Gender:
Posts: 585
|
|
Re: An angle in a triangle
« Reply #6 on: Jan 11th, 2013, 8:44am » |
Quote Modify
|
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.
|
|
IP Logged |
|
|
|
jollytall
Senior Riddler
Gender:
Posts: 585
|
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°.
|
|
IP Logged |
|
|
|
Mariko79
Newbie
Better than Van Damme
Posts: 30
|
|
Re: An angle in a triangle
« Reply #9 on: Apr 8th, 2013, 11:25pm » |
Quote Modify
|
on Jan 10th, 2013, 10:58am, 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... .
|
|
IP Logged |
|
|
|
Rosiethomas
Newbie
Posts: 4
|
|
Re: An angle in a triangle
« Reply #10 on: Jul 1st, 2013, 3:29am » |
Quote Modify
|
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-triang le.aspx
|
|
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print
Construction.pdf