W------------------------X
       |                      * |
       |    A              *    |          
       |                O       |
       |             *  * *  B  |	 
       |          *     *  *    |
       |       *        *   *   |
       |    *           *    *  |
       | *      D       *  C  * |
       Z----------------Q-------Y

What is the minimum area of rectangle WXYZ if all lengths are whole numbers, as are the areas of the similar triangles, denoted by A, B, C & D?

Let i, j, and k be integers with k^2 = i^2 + j^2.

The sides of the triangles are

  Triangle  k-side  i-side  j-side    Area

    XZW     (kk)k   (kk)i   (kk)j     ij(k^4)/2

    ZOQ     (jj)k   (jj)i   (jj)j     i(j^5)/2

    YXO     (ik)k   (ik)i   (ik)j     (i^3)j(k^2)/2

    OYQ     (ij)k   (ij)i   (ij)j     (i^3)(j^3)/2

The area of the rectangle is ij(k^4)

If (i,j,k) = (3,4,5), we get the minimum area

The sides of the triangles are

  Triangle  k-side  i-side  j-side    Area

    XZW      125      75    100       3750

    ZOQ       80      48     64       1536

    YXO       75      45     60       1350

    OYQ       60      36     48        864

The area of the rectangle is 7500.

Posted by Bractals on 2008-03-18 10:45:53