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Thread: Area of part of rectangle made out of diagonals

  1. #1
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    Area of part of rectangle made out of diagonals

    I have attached an image.

    E is in the middle between B and C.

    The question is: Is the rectangle ABHG a third of the whole rectangle ABCD?

    I have tried to find some uniform triangles but I don't know what to do with them. Please help!
    Attached Thumbnails Attached Thumbnails Area of part of rectangle made out of diagonals-geometry.jpg  
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    Re: Area of part of rectangle made out of diagonals

    Hey EulerWannabe.

    For this kind of problem it's usually a good idea to state all of the mathematical relationships that a particular object has.

    Can you show us your working in regard to this problem?
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    Re: Area of part of rectangle made out of diagonals

    I found some angles to be equal in size:
    Attached Thumbnails Attached Thumbnails Area of part of rectangle made out of diagonals-angles.jpg  
    Last edited by EulerWannabe; Apr 17th 2017 at 03:28 AM.
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    Re: Area of part of rectangle made out of diagonals

    The easiest solution is to assign coordinates to the rectangle:

    Thanks from HallsofIvy
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    Re: Area of part of rectangle made out of diagonals

    Quote Originally Posted by EulerWannabe View Post
    I found some angles to be equal in size:
    Triangles BFE and DFA are similar
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    Re: Area of part of rectangle made out of diagonals

    Quote Originally Posted by johng View Post
    The easiest solution is to assign coordinates to the rectangle:

    I don't know if the "easiest solution is to assign coordinates to the rectangle." But it's easier than you have it if you:

    1) Imagine the height of the rectangle having a height of 1, because the height is relatively irrelevant for the sake of this problem.

    2) Because E is midway between B and C, call E "a" instead of "a/2." And call C "2a" to avoid fractions until nearer the end.


    So, let A = (0, 0), B = (0, 1), C = (2a, 1), D = (2a, 0), E = (a, 1).

    Just find the x-coordinate of F.

    You would have some similar work as you have shown to the right, but you would have fewer variables and fewer fractions.

    And the area of the smaller (desired) rectangle equals the x-coordinate of F (which is in terms of a), divided by the length of the largest rectangle (which is 2a).
    Last edited by greg1313; Apr 18th 2017 at 02:08 PM.
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