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anonymous

  • one year ago

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  1. ganeshie8
    • one year ago
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    |dw:1439559062484:dw|

  2. anonymous
    • one year ago
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    not all of them will pass through the center though...

  3. ganeshie8
    • one year ago
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    Notice that the center of circle is the "circumcenter" of the triangle we're working on

  4. ganeshie8
    • one year ago
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    so basically the problem is equivalent to finding the triangles such that the circumcenter lies interior to the triangle

  5. anonymous
    • one year ago
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    oh, yes

  6. ganeshie8
    • one year ago
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    for what triangles do we have circumcenter interior to the triangle ?

  7. anonymous
    • one year ago
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    acute triangles

  8. ganeshie8
    • one year ago
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    yes, they would be all "acute" triangles so our job is to find the number of acute triangles and divide them by 84

  9. ganeshie8
    • one year ago
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    we may use the relation between inscribed angle and central angle : \(\theta = \dfrac{\alpha}{2}\) |dw:1439560115696:dw|

  10. ganeshie8
    • one year ago
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    not really sure how to approach this, im still thinking..

  11. anonymous
    • one year ago
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    Thanks so much for helping me, sir. But I must go to sleep now, its getting late for me. I will think about this tomorrow.

  12. ganeshie8
    • one year ago
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    Okay, have good sleep :) I'll try to post the solution over the night, pretty sure this is not that hard...

  13. anonymous
    • one year ago
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    thanks so much! i really appreciate it!

  14. Loser66
    • one year ago
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    |dw:1439562522979:dw|

  15. Loser66
    • one year ago
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    Consider vertex A. From AB, we have 7 triangles, and among them just ABF has center inside of it. That is the probability to get the center inside of the triangle is 1/7 for vertex A Same for others Hence the total is 1/7^9 but we have to subtract the overlap parts. I meant \(\triangle ABC\) when consider node A will be overlap with \(\triangle BCA\) for node B.

  16. Loser66
    • one year ago
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    Hence for node B, we have 1 triangle overlaps with node A for node C, we have 1triangles overlaps with node A \(\triangle ACD\), and 1 triangle overlaps with nod B \(\triangle CBA\) Same for other nodes and same argument, we have the logic 1st node --0 overlap 2nd node--1 overlap 3rd node---2 overlap ::::::::::::::::::::::::::::::: 9th node---8 overlap ------------------------ total 36 cases.

  17. Loser66
    • one year ago
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    |dw:1439564084039:dw|

  18. Loser66
    • one year ago
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    |dw:1439564259300:dw|

  19. Loser66
    • one year ago
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    Ok, part 1) is done, now just calculate the possibility of it.

  20. ganeshie8
    • one year ago
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    for part a, im getting \(\dfrac{30}{84}\) for partb, the general formula for probability is \(\dfrac{n+1}{2(2n-1)}\)

  21. anonymous
    • one year ago
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    can you tell me how, i need to write an explanation with my answer...

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