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Triangulation of Numbers (Posted on 2008-03-18) Difficulty: 3 of 5
       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?

See The Solution Submitted by brianjn    
Rating: 3.6667 (3 votes)

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Solution Solution | Comment 1 of 6

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
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