All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
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)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution | Comment 2 of 6 |

The guess is that each triangle is similar to a 3,4,5 right triangle.

If QY were 3, QO were 4 and OY 5, then OX would be 3*5/4=15/4, and YX would be 25/4, and ZX would be 125/12.

So if we scale that up by a factor of 12, everything would be integral, with ZX at 125 units and XY at 75 and ZY at 100, making the area of WXYZ = 7500.

I don't have a proof that this is a minimum, however, so the following program checks for solutions with an area of 7500 or less.

DEFDBL A-Z
FOR h = 1 TO SQR(7500)
  FOR w = h + 1 TO 7500 / h
    sumSq = h * h + w * w
    diag = INT(SQR(sumSq) + .5)
    IF diag * diag = sumSq THEN
       OY = h * w / diag
       OX = h * h / diag
       IF OY = INT(OY) AND OX = INT(OX) THEN
         OQ = OY * w / diag
         QY = OY * h / diag
         IF OQ = INT(OQ) AND QY = INT(QY) THEN
           PRINT h, w, diag
           PRINT OY, OX, OQ, QY
           PRINT h * w, h * w / 2, OX * OY / 2, QY * OQ / 2, OQ * (w - QY) / 2
           PRINT
         END IF
       END IF
    END IF
  NEXT w
NEXT h

where h and w are the height and width of the large rectangle.

In the above, diag represents the length of ZX, and OY = h * w / diag since

OY/XY = ZY/ZX due to triangle B's similarity to triangle A.

Similarly for OX, as OX/YX = YX/ZX.

Then OQ/OY = ZY/ZX and QY/OY = YX/ZX are used to find OQ and QY.

If everything is integral, the height, width and diagonal are shown, followed on the next line by OY, OX, OQ and QY. As the difference between integers is also an integer, we don't need to verify ZQ and OZ. The last line shows the area: h*w, and the area A, area B, area C and area D. The result is:

 ZW=YX=75      WX=ZY=100     ZX=125
 OY=60         OX=45         OQ=48         QY=36
 Area of WXYZ=7500  Area A=3750  Area B=1350  Area C=864  Area D=1536
 

So there is nothing smaller than the initial guess.

 


  Posted by Charlie on 2008-03-18 10:50:27
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (7)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information