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Thread: Maths Question!

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    Question Maths Question!

    Hi,

    I've tried solving this question a few times, however I can't seem to do and I think the points that have been provided are incorrect. Here is the question:

    The points A(-3,4) B(-5,-2) and C(2,6) form the vertices of a triangle. Prove that the perpendicular bisectors of the vertices are concurrent.

    Could someone please provide the solution?

    Thanks a lot for the help!

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    Re: Maths Question!

    Quote Originally Posted by OriginalCopy View Post
    Hi,

    I've tried solving this question a few times, however I can't seem to do and I think the points that have been provided are incorrect. Here is the question:

    The points A(-3,4) B(-5,-2) and C(2,6) form the vertices of a triangle. Prove that the perpendicular bisectors of the vertices are concurrent.

    Could someone please provide the solution?

    Thanks a lot for the help!
    Doesn't make sense. You cannot bisect a vertex!

    But the perpendicular bisectors of the 3 sides of any proper triangle on a plane are concurrent. For this specific case, if P, Q and R are the mid-points of the 3 sides BC, CA and AB: you can find their mid-points P, Q and R, the gradients of BC, CA and AB, and therefore of their perpendiculars. Then you can find the equations of these 3 perpendiculars. Find where 2 perps intersect; show this point of intersection lies on the 3rd perpendicular.
    Last edited by Drongoski; 9 Oct 2017 at 7:58 PM.
    OriginalCopy likes this.
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    Re: Maths Question!

    Thanks! I did try to do that method, however the solution I got showed that the perpendicular bisectors were not concurrent.

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    Re: Maths Question!

    Quote Originally Posted by OriginalCopy View Post
    Thanks! I did try to do that method, however the solution I got showed that the perpendicular bisectors were not concurrent.
    The 3 mid-points are:


    and the gradients of the perpendiculars at P, Q and R are resp: -7/8, -5/2 and -1/3

    The equations of the 3 perpendiculars are (not very nice numbers):

    P: y = -7x/8 + 11/16
    Q: y = -5x/2 + 15/4
    R: y = -x/3 - 1/3

    The first 2 intersect at: (49/26, -0.96153 ...)
    You now use x = 49/26 in the 3rd equation; should get y = -0.96153 ...

    .: the 3 perpendicular are concurrent.

    But this concurrency holds for all plane triangles; not just this particular case.
    Last edited by Drongoski; 11 Oct 2017 at 8:30 PM.
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