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Math Help - Proving that each diagonal of a parallelogram bisects each other

  1. #1
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    Proving that each diagonal of a parallelogram bisects each other

    Q: Prove that each diagonal of a parallelogram bisects each other

    How do I attempt this?

    Can I find the midpoints of the diagonals, then if they're the same, get the distance between this midpoint and the vertices? If they're the same, have I proved it?

    Thanks.
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  2. #2
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    Quote Originally Posted by Glitch View Post
    Q: Prove that each diagonal of a parallelogram bisects each other

    How do I attempt this?

    Can I find the midpoints of the diagonals, then if they're the same, get the distance between this midpoint and the vertices? If they're the same, have I proved it?

    Thanks.
    It can be proved using vectors. Consider a parallelogram ABCD with a,b,c,d as the position vectors of points A,B,C,D respectively.

    \vec{AB}=\vec{DC}

    so b-a=c-d

    Rearrange it to get a+c=b+d

    Divide both sides by 2, (a+c)/2=(b+d)/2

    We see that the midpoints of AC and BD are identical.

    ...
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  3. #3
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    I don't think I'm supposed to solve this with vectors, since I haven't learnt about them yet.
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  4. #4
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    use congruent triangles, then.
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  5. #5
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    Quote Originally Posted by Glitch View Post
    I don't think I'm supposed to solve this with vectors, since I haven't learnt about them yet.
    There's the problem- if you don't show any work at all, we have no idea what you can use!

    In addition to "congruent triangles" (basic geometry) and "vectors", I might use "coordinate geometry". We can always set up a coordinate system so that one vertex is at (0, 0), another at (a, 0), a third at (b, c) and then the fourth must be at (a+ b, c). Now, it is easy to find the midpoint of each diagonal- that is, if you can use coordinate geometry- and we don't know that.
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  6. #6
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    Quote Originally Posted by HallsofIvy View Post
    There's the problem- if you don't show any work at all, we have no idea what you can use!
    I just wanted to know if my reasoning was sound.
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