Acute angled circular
- Raiyan Jamil
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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 .
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Re: Acute angled circular
i think ans is $C(7,3)-7=28$. is my ans correct???
Re: Acute angled circular
Give an explanation if possible.
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- Fatin Farhan
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Re: Acute angled circular
There are $$C(7,3)-7-14=14$$ acute angled triangle.*Mahi* wrote:Yes it is, but give an explanation if possible.
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 $$14$$
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- Raiyan Jamil
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Re: Acute angled circular
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 .
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 .
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- Fatin Farhan
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Re: Acute angled circular
If you kindly explain what you are saying. The number of all triangles is not \(343\), it's \(35\)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 .
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- asif e elahi
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Re: Acute angled circular
The number of triangles is $\binom{7}{3}=35$
- Raiyan Jamil
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Re: Acute angled circular
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 ?
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 ?
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- nishat protyasha
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Re: Acute angled circular
@ 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.
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.
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Re: Acute angled circular
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 $\frac{7 \times 6}{3}=14$.
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