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Math Help - Find number of different triangles possible?

  1. #1
    Super Member fardeen_gen's Avatar
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    Find number of different triangles possible?

    The sides of a triangle are a, b and c where a, b and c are integers and a\leq b\leq c. If c is given, show that the number of different triangles is \frac{1}{4}c(c + 2) or \frac{1}{4}(c + 1)^2, according as c is even or odd. Also, show that the number of isoceles or equilateral triangle is \frac{1}{2}(3c - 2) or \frac{1}{2}(3c - 1), according as c is even or odd.
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  2. #2
    Junior Member
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    The important fact here is that in a triangle with sides a,b,c, it must be true that a+b>c, a+c>b, and b+c>a.

    The last two inequalities are obvious given that a\leq b\leq c, so we just need to see how many ways we can choose a and b so that a+b>c.

    Suppose c is even.

    Then if a=1, then b must equal c.
    if a=2, then b can equal c or c-1.
    ..
    if a=c/2, then b can equal c/2+1, ... , c.

    (So far we have 1+2+\cdots+c/2=\frac{1}{8}c(c+2) ways to choose a and b.

    Doing the same thing when a>c/2, we find that there are \frac{1}{8}n(n+2) more ways to choose, giving a total of \frac{1}{4}c(c+2) triangles.

    The other cases of this problem are not much different.
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