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Math Help - Trapezium Proof

  1. #1
    Member jacs's Avatar
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    [SOLVED with many thanks] Trapezium Proof

    ABCD is a trapezium, with AB||CD. A is joined to C and B is joined to D. Point E is the midpoint of BD and point F is the midpoint of AC. E is joined to F.
    Show that the line EF is half the distance of the difference between DC and AB.



    Not sure if this is sound but was how I started:
    If E and and F are midpoints of BD and AC respectively, and AB||CD then EF must be parallel to AB and CD (family of parallel lines cut off intercepts in equal ratios)

    then i thought that I could use similar triangles. Letting diagonals intersect at X and after proving \DeltaXEF similar \DeltaXDC and \DeltaAXB with sides in equal ratios, but can't figure out how to get it structured right or proceed from there.

    Been at it a while now and it perhaps has become a issue of can't see the forest for the trees, but am defnintely hittng a brick wall.

    any help greatly appreciated
    Attached Thumbnails Attached Thumbnails Trapezium Proof-trapezium.png  
    Last edited by jacs; January 16th 2011 at 05:19 PM. Reason: to mark as solved
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  2. #2
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    Quote Originally Posted by jacs View Post
    ABCD is a trapezium, with AB||CD. A is joined to C and B is joined to D. Point E is the midpoint of BD and point F is the midpoint of AC. E is joined to F.
    Show that the line EF is half the distance of the difference between DC and AB.



    Not sure if this is sound but was how I started:
    If E and and F are midpoints of BD and AC respectively, and AB||CD then EF must be parallel to AB and CD (family of parallel lines cut off intercepts in equal ratios)

    then i thought that I could use similar triangles. Letting diagonals intersect at X and after proving \DeltaXEF similar \DeltaXDC and \DeltaAXB with sides in equal ratios, but can't figure out how to get it structured right or proceed from there.

    Been at it a while now and it perhaps has become a issue of can't see the forest for the trees, but am defnintely hittng a brick wall.

    any help greatly appreciated
    Extend the line [AB] out to the right and left, forming a rectangle.

    Then extend [EF] until the line meets [BC] at G.

    Then it is easy to see that |EG|=0.5|DC| and |FG|=0.5 |AB|

    from which the result follows.

    (There's really no need to draw the rectangle, it may help make the picture a little clearer).
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  3. #3
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by jacs View Post
    ABCD is a trapezium, with AB||CD. A is joined to C and B is joined to D. Point E is the midpoint of BD and point F is the midpoint of AC. E is joined to F.
    Show that the line EF is half the distance of the difference between DC and AB.



    Not sure if this is sound but was how I started:
    If E and and F are midpoints of BD and AC respectively, and AB||CD then EF must be parallel to AB and CD (family of parallel lines cut off intercepts in equal ratios)

    then i thought that I could use similar triangles. Letting diagonals intersect at X and after proving \DeltaXEF similar \DeltaXDC and \DeltaAXB with sides in equal ratios, but can't figure out how to get it structured right or proceed from there.

    Been at it a while now and it perhaps has become a issue of can't see the forest for the trees, but am defnintely hittng a brick wall.

    any help greatly appreciated
    join AE and let AE meet DC at G.
    then triangles AEB and DEF are congruent. Hence AE=EG.
    Then in triangle AGC observe that E is mid-point of AG and F is mid-point of AC. now use mid-point theorem(or triangle mid-line theorem) to get EF=(1/2)*(DC-DG)
    but DG=AB hence EF=(1/2)(DC-AB).
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  4. #4
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    Trapezium Proof-untitled.gifLook at the extended diagram .
    Note that 2m(\overline{PQ})= m(\overline{AB})+ m(\overline{CD}) .

    Also note that 2m(\overline{FP})= m(\overline{DC}).

    If we continue this way, then the result will follow.
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  5. #5
    Member jacs's Avatar
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    Thank you all so very much, this has been a massive help and it is awesome to see so many different approaches. This has been incredibly helpful and has given me a few ideas to think 'outside the box' or trapezium, in this case. Really appreciate your help!

    Thanks again
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