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Math Help - How to find the largest square inside a triangle?

  1. #1
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    How to find the largest square inside a triangle?

    I am wondering how you would go about to find the largest square that would fit inside a triangle?

    An example of what i am asking is:

    What is the side length of the largest square that would fit inside a right angled triangle with the sides 5,12, and 13?

    Any help would be appreciated

    Medusa
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  2. #2
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    The largest rectangle that would fit inside a right triangle with the sides 5, 12, 13 is a square. Let the square have side x. By Pythagorean theorem, (12\,-\,x)^2\,+\,x^2\,=\,(13\,-\,\sqrt{2x^2\,-\,10x\,+\,25})^2. Solving the equation gives x = 60/17.
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  3. #3
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    Thank you for the help. I trying to use this method but i kept stuffing up the second part. You have made it a lot clearer now.

    Thanks

    Medusa
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  4. #4
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    After doing further research into this question and many discussions with my math genius friends i was wondering if anyone could explain how the formula: bh/b+h works in this situation? b = base, h = height. I am unsure how this works and any help explaining it would be appreciated.
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  5. #5
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    Hello, Medusa!

    Can anyone explain how the formula: bh/(b+h) works in this situation?
    b = base, h = height.
    Code:
    -  - Ao
    :  :  | *
    : h-x |   *
    :  :  |     *
    :  - Do-------oE
    h  :  |   x   | *
    :  :  |       |   *
    :  x  |      x|     *
    :  :  |       |       *
    :  :  |       |          *
    -  -  o-------o-------------o
          B - x - F - - b-x - - C
          : - - - -  b  - - - - :

    We have right triangle ABC with AB = h, BC = b.

    We have square BDEF with sides x.


    Since ∆ABC ~ ∆ADE, we have:

    . . .h. . . .h - x
    . . --- .= .----- . . . . hx .= .b(h - x) . . . . hx .= .bh - bx
    . . .b. . . . . x

    . . bx + hx .= .bh . . . . (b + h)x .= .bh

    . . . . . . . . . . . . . . .bh
    Therefore: . x .= .-------
    . . . . . . . . . . . . . .b + h

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  6. #6
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    So the formula would work for a right angled triangle.. What about if the triangle was isoceles? Would the formula still work if the triangle isn't right angled is pretty much what i am asking.

    Medusa
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  7. #7
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    Quote Originally Posted by Medusa View Post
    So the formula would work for a right angled triangle.. What about if the triangle was isoceles? Would the formula still work if the triangle isn't right angled is pretty much what i am asking.

    Medusa
    1. Draw a sketch. see attachment

    2. Use similar triangles. You'll get the proportions:

    Attached Files Attached Files
    Thanks from chihuahuab
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  8. #8
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    Thanks for the help I sat there and worked it out using your advice and it makes sense after completely working it out by hand Thanks for help

    Medusa
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  9. #9
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    Re: How to find the largest square inside a triangle?

    Quote Originally Posted by johnny View Post
    The largest rectangle that would fit inside a right triangle with the sides 5, 12, 13 is a square. Let the square have side x. By Pythagorean theorem, (12\,-\,x)^2\,+\,x^2\,=\,(13\,-\,\sqrt{2x^2\,-\,10x\,+\,25})^2. Solving the equation gives x = 60/17.
    Or simply (a=5, b=12) x = ab / (a + b) ; (product of legs) / (sum of legs)
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