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Math Help - Parallelogram

  1. #1
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    Parallelogram

    In the figure, ABCD is a parallelogram.
    If DX=XY=YB then

    (a)AX||YC
    (b)XAYC is a parallelogram
    (c)Ac bisects XY
    (d)All of these.
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by devi View Post
    In the figure, ABCD is a parallelogram.
    If DX=XY=YB then

    (a)AX||YC
    (b)XAYC is a parallelogram
    (c)Ac bisects XY
    (d)All of these.
    I'm only going to give you tips because it's quite a hard work proving all of these ^^
    For those who don't see the sketch, D, X, Y and B are aligned on the diagonal DB.

    DX=XY=YB. Prove that DX=DB/3 and BY=DB/3 (1)

    (a) Prove that triangles AXB and DCY are isometric (2 equal sides and the angle inscribed equal). Then, note that 2 of their sides are parallel to each other and juggle with angles.

    (b) This is a direct consequence from (a) : AX // YC and AX=YC. This is a sufficient property to prove that it's a parallelogram.

    (c) Property (1) + property of the parallelogram : its diagonals intersect in their midpoint (let's call it O for parallelogram ABCD). This may be sufficient to prove that XO=YO and thus that AC bisects XY.

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