Acute angled circular

For students of class 6-8 (age 12 to 14)
User avatar
Raiyan Jamil
Posts:138
Joined:Fri Mar 29, 2013 3:49 pm
Acute angled circular

Unread post by Raiyan Jamil » Sat Feb 08, 2014 8:51 pm

In a circle , 7 points are marked on it so that the distance between every two consecutive points are the same . Now , at maximum , how many acute angled triangles could be made by joining any three points ? Also remember that , sometimes a type of triangle could be uncountable for its not being a acute angled triangle .
A smile is the best way to get through a tough situation, even if it's a fake smile.

Tahmid
Posts:110
Joined:Wed Mar 20, 2013 10:50 pm

Re: Acute angled circular

Unread post by Tahmid » Sun Feb 09, 2014 12:25 am

i think ans is $C(7,3)-7=28$. is my ans correct??? :?:

User avatar
*Mahi*
Posts:1175
Joined:Wed Dec 29, 2010 12:46 pm
Location:23.786228,90.354974
Contact:

Re: Acute angled circular

Unread post by *Mahi* » Sun Feb 09, 2014 7:01 am

Give an explanation if possible.

[Sorry, didn't check! :P Thanks Fatin!]
Please read Forum Guide and Rules before you post.

Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

User avatar
Fatin Farhan
Posts:75
Joined:Sun Mar 17, 2013 5:19 pm
Location:Kushtia,Bangladesh.
Contact:

Re: Acute angled circular

Unread post by Fatin Farhan » Sun Feb 09, 2014 9:46 am

*Mahi* wrote:Yes it is, but give an explanation if possible.
There are $$C(7,3)-7-14=14$$ acute angled triangle.
If we count the obtuse angled triangles then we have 2 case. First place points $$A,B,C,D,E,F,G$$. Then we have 2 types of obtuse angled triangle: $$ABC$$ type and $$ABD$$ type.
We have $$7$$ triangles of $$ABC$$ type and $$14$$ triangles of $$ABD$$ type. So, the number of acute angled triangle is $$C(7,3)-7-14=14$$.
Also if we count the acute angle triangles then we have 2 cases:
$$ABE$$ and $$ACE$$ type. There are 7 $$ABE$$ type triangles and 7 $$ACE$$ type triangles. So total $$14$$
"The box said 'Requires Windows XP or better'. So I installed L$$i$$nux...:p"

User avatar
Raiyan Jamil
Posts:138
Joined:Fri Mar 29, 2013 3:49 pm

Re: Acute angled circular

Unread post by Raiyan Jamil » Sun Feb 09, 2014 2:48 pm

Raiyan speaking ,
I don't understand , if the points are A , B , C , D , E , F and G . Everyone says 14 or 28 are possible . But taking all types of triangle , we get 343 types . So , there might be much more types . Please try to think the ans carefully .
A smile is the best way to get through a tough situation, even if it's a fake smile.

User avatar
Fatin Farhan
Posts:75
Joined:Sun Mar 17, 2013 5:19 pm
Location:Kushtia,Bangladesh.
Contact:

Re: Acute angled circular

Unread post by Fatin Farhan » Sun Feb 09, 2014 3:14 pm

Raiyan Jamil wrote:Raiyan speaking ,
I don't understand , if the points are A , B , C , D , E , F and G . Everyone says 14 or 28 are possible . But taking all types of triangle , we get 343 types . So , there might be much more types . Please try to think the ans carefully .
If you kindly explain what you are saying. The number of all triangles is not \(343\), it's \(35\)
"The box said 'Requires Windows XP or better'. So I installed L$$i$$nux...:p"

User avatar
asif e elahi
Posts:185
Joined:Mon Aug 05, 2013 12:36 pm
Location:Sylhet,Bangladesh

Re: Acute angled circular

Unread post by asif e elahi » Sun Feb 09, 2014 5:35 pm

The number of triangles is $\binom{7}{3}=35$

User avatar
Raiyan Jamil
Posts:138
Joined:Fri Mar 29, 2013 3:49 pm

Re: Acute angled circular

Unread post by Raiyan Jamil » Sun Feb 09, 2014 8:36 pm

Raiyan speaking ,
Sorry for that , I have heard it from my teacher may'be . Or, I am mistaking . I forgot that to make a triangle , a point could not be made by taking two or three times . So I mistakenly did 7 cube 0r 343 . Sorry for that but I still don't understand why 35 as I am in Junior catagory of the matholympiad . I read it but I have forgotten . Please tell me why 35 happened ?
A smile is the best way to get through a tough situation, even if it's a fake smile.

User avatar
nishat protyasha
Posts:33
Joined:Tue Sep 17, 2013 12:02 am
Location:Sylhet, Bangladesh.

Re: Acute angled circular

Unread post by nishat protyasha » Mon Feb 10, 2014 9:48 pm

@ Raiyan
When we want to make a triangle we have to choose three points. That's why we will choose 3 points from 7 points. So C(7,3) will be the number of total triangles.

Nayeemul Islam Swad
Posts:22
Joined:Sat Dec 14, 2013 3:28 pm

Re: Acute angled circular

Unread post by Nayeemul Islam Swad » Tue Feb 11, 2014 9:01 pm

As the distance between every two consecutive points are the same, any triange is acute-angled iff the difference between the points is $\leq 3$. Now, there are $7$ points. So the number of triangles having that property is one-third of the coefficient of the term $x^7$ multiplied by $7($as each point has been counted thrice$)$ in the expansion of $(x+x^{2}+x^{3})^{3}.$ After applying the method of synthetic multiplication we find that the coefficient of $x^{7}$ is $6$. So the number of such triangles is $\frac{7 \times 6}{3}=14$.
Why so SERIOUS?!??!

Post Reply