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?

The smallest right triangle with integer sides is a triangle with a short leg of 3, a long leg of 4, and a hypotenuse of 5. As all the five triangles formed are similar, and to keep to the smallest area, we can use these numbers as our factors for the sides of the triangles that are formed.

QO is OQY triangle's long leg and OQZ triangle's short leg.
OY is OQY triangle's hypotenuse and OXY triangle's long leg.
XY is OXY triangle's hypotenuse and equal in length to WZ, WXZ triangle's short leg.


QY = 36 = (3 * 4 * 3)   QO = 48 = (3 * 4 * 4)
QO = 48 = (4 * 4 * 3)   QZ = 64 = (4 * 4 * 4)
YO = 60 = (5 * 4 * 3)   OZ = 80 = (5 * 4 * 4)

OX = 45 = (3 * 3 * 5)   WZ = 75 = (3 * 5 * 5)
OY = 60 = (4 * 3 * 5)   WX =100 = (4 * 5 * 5)
XY = 75 = (5 * 3 * 5)   ZX =125 = (5 * 5 * 5)


W------------------------X
|                      * |
|                   *    |
|                O       |
|             *  * *     |
|          *     *  *    |
|       *        *   *   |
|    *           *    *  |
| *              *     * |
Z----------------Q-------Y

The area of the newly formed rectangle, WXYZ, calculated from the lengths of the two sides (XY and WX) is, therefore, 75 * 100 = 7500 sq. units.

Posted by Dej Mar on 2008-03-19 00:08:05