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Prove that the greatest angle is 120

  1. Jan 18, 2013 #1

    utkarshakash

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    1. The problem statement, all variables and given/known data
    The sides of a triangle are [itex]x^2+3x+3, 2x+3,x^2+2x[/itex]. Prove that the greatest angle of the triangle is 120°.

    2. Relevant equations

    3. The attempt at a solution
    The greatest angle is the one opposite to the greatest side. But how to decide which one is greatest side? Also the sides are variable.
     
  2. jcsd
  3. Jan 19, 2013 #2

    ehild

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    The side lengths must be positive numbers. Is x2+3x+3>x2+2x? Is x2+3x+3>2x+3?

    ehild
     
  4. Jan 19, 2013 #3

    haruspex

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    Assuming x > 0, it should be obvious which is the longest side. (But you can't say in general which is the shortest.)
    Do you know the cosine rule relating lengths of sides of a triangle?
     
  5. Jan 19, 2013 #4

    Mark44

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    It depends on the value of x. If you graph y1 = x^2 + 3x + 3, y2 = x^2 + 2x, and y3 = 2x + 3, two of the graphs are parabolas and one is a straight line. On the interval [0, 1], the largest values are for y1 = x^2 + 3x + 3 and the smallest are for y2 = x^2 + 2x. For the interval [-1, 0] the order is different.
     
  6. Jan 19, 2013 #5

    haruspex

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    No, it turns out not to.
     
  7. Jan 19, 2013 #6

    Mark44

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    By what I said, I meant that the values of the three expressions depend on x. As you said, one of the sides is the longest, but for other two sides, it depends on which interval you're looking at.
     
  8. Jan 19, 2013 #7

    I like Serena

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    Turns out x has to be positive, otherwise one of the sides has a negative length.
    And if x is positive, there is only one side that can be longest.

    I found the results were even more impressive when I worked it through...
     
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