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Math Help - triangle area proof

  1. #1
    Member jacs's Avatar
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    triangle area proof

    In triangle ABC, X is the midpoint of AB and Y is the midpont of AC.
    XC is joined. Prove
    a) ∆AXY = ∆XYC in area
    b) ∆AXC = half ∆ABC in area
    c) Find what fraction the area of ∆AXY is of the ∆ABC


    thanks


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  2. #2
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    earboth's Avatar
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    Quote Originally Posted by jacs
    In triangle ABC, X is the midpoint of AB and Y is the midpont of AC.
    XC is joined. Prove
    a) ∆AXY = ∆XYC in area
    b) ∆AXC = half ∆ABC in area
    c) Find what fraction the area of ∆AXY is of the ∆ABC


    thanks


    Hello,

    to a). Both triangles have the same base: XY and the same height, because XY divides the height of ABC in 2 equal parts.

    to b) XC divides (ABC) in 2 equal areas.

    to c) Because area(AXY) = area(XYC) and area(ABC) = 2 * area(AXC) therefore area(ABC) = 4 * area(AXY)

    (The three midparallels of a triangle divide the triangle in 4 congruent triangles, similar to the great one)

    I've attached a copy of your diagram to demonstrate how I calculated.

    Greetings

    EB



    to b)
    Attached Thumbnails Attached Thumbnails triangle area proof-teilfl_3eck.gif  
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  3. #3
    Member jacs's Avatar
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    thanks for that EB, i see it now and i should have seen it before (duh to me) I actually cheated and used trig area of triangle rule for part a, utilising
    sin a = sin (180 - a) which i knew was not the way to go but couldnt see the other way.

    thanks for that

    jacs
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