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Math Help - Triangle vertex

  1. #1
    Senior Member euclid2's Avatar
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    Triangle vertex

    The points A(5,1) and B(-3,6) represent one of the equal sides of an isosceles triangle. Determine one of the possible points that would represent the third vertex of the triangle. Provide calculations to support your answer.

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    Quote Originally Posted by euclid2 View Post
    The points A(5,1) and B(-3,6) represent one of the equal sides of an isosceles triangle. Determine one of the possible points that would represent the third vertex of the triangle. Provide calculations to support your answer.

    Thanks
    The third vertex of the triangle is located on a circle around A with radius r = \sqrt{89} , that means the circle passes through B.

    OR

    the third vertex of the triangle is located on a circle around B with radius r = \sqrt{89} , that means the circle passes through A.

    EDIT: I've repaired a silly mistake. Thank you, kalagota!
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    MHF Contributor kalagota's Avatar
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    first, compute the distance between A and B.

    second, the set of all points that satisfies you condition is the set of (x,y) such that the distance between A (or B) and (x,y) is equal to the distance between A and B.

    so for example, d^2 = (x_B-x_A)^2 + (y_B-y_A)^2 = 89
    we consider: from point A, we will find a point (x,y) such that the distance of A and (x,y) is d.

    so d^2 = 89 = (x-5)^2 + (y-1)^2 (this equation represents a circle) the point is, you can find any (x,y) that satisfies that equation..

    NOTE: if you find an (x,y), be sure that A,B and (x,y) are not collinear.

    you can also do this with respect to point B. so, there will be a new equation.. with same d.
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    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by earboth View Post
    The third vertex of the triangle is located on a circle around A with radius r = \sqrt{91} , that means the circle passes through B.

    OR

    the third vertex of the triangle is located on a circle around B with radius r = \sqrt{91} , that means the circle passes through A.
    91? hmm, (5 - -3)^2 + (1 - 6)^2 = 8^2 + (-5)^2 = 64 + 25
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    In addition to my previous post I've attached a sketch of the situation.

    I've choosen a point on the circle c_A (approximately (-8, -2) check if this point lies exactly on this circle!)

    and

    a point on the circle c_B (approximately (0,-7) check if this point lies exactly on this circle!)
    Attached Thumbnails Attached Thumbnails Triangle vertex-glschenklgdreieck_inkrsn.gif  
    Last edited by earboth; July 10th 2008 at 10:07 PM.
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  6. #6
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by earboth View Post
    a point on the circle c_B (approximately (-7, 0) check if this point lies exactly on this circle!)
    that should be (0,-7)
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    Senior Member euclid2's Avatar
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    are you suggesting that either one of those points are appropriate for the third vertex, if so how did you come up with those? and how do i verify they are correct?
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  8. #8
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    Quote Originally Posted by euclid2 View Post
    are you suggesting that either one of those points are appropriate for the third vertex, if so how did you come up with those? and how do i verify they are correct?
    You only have to use the equations kalagota has posted in this thread:

    Take (-8, -2) and plug in the coordinates into the appropriate equation:

    (x+3)^2+(y-6)^2=89 will yield:

    (-8+3)^2+(-2-6)^2=89 Check! So (-8,-2) is indeed a vertex of the isosceles triangle.
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  9. #9
    Senior Member euclid2's Avatar
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    Ok thanks.
    I'm just not sure where you get the (-8,-2) from that's all?
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  10. #10
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    Quote Originally Posted by euclid2 View Post
    Ok thanks.
    I'm just not sure where you get the (-8,-2) from that's all?
    If you have a look at the drawing I posted in one of my previous post you'll probably find some points with integer coordinates which lie on one of the circles. These points are my first guess.

    Since a drawing isn't accurate enough I advised you to check, if the point lies exactly on the circle by using the equations of the circle. That's all, nothing mysterious about it.
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