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Weighted Quadrilateral (Posted on 2013-07-25) Difficulty: 2 of 5
A 3x4 rectangle has sides 3,4,3,4 and diagonals 5,5. Its "weight" is said to be 3+4+3+4+5+5 = 24. Find a quadrilateral with all sides and diagonals having integer length and whose weight is smaller than 24.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 3.0000 (1 votes)

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Solution computer solution Comment 1 of 1

The program considers two back-to-back triangles sharing a segment that forms one of the diagonals of the quadrilateral:

DECLARE FUNCTION cvdg# (x#)
DECLARE FUNCTION angle1# (a#, b#, c#)
DEFDBL A-Z
DIM SHARED pi
pi = ATN(1) * 4
CLS

FOR t = 5 TO 24
  FOR a = 1 TO t / 2
    b = t - a
    FOR d1 = b - a + 1 TO a + b - 1
         angleA = angle1(a, b, d1)
         angleB = angle1(b, a, d1)
         FOR t2 = d1 + 1 TO 24
            FOR c = 1 TO t2 - 1
              d = t2 - c
              IF d1 + c > d AND d1 + d > c THEN
                 angleC = angle1(c, d, d1)
                 angleD = angle1(d, c, d1)
                 d2sq = b * b + c * c - 2 * b * c * COS(angleA + angleD)
                 d2 = SQR(d2sq)
                 d2rnd = INT(d2 + .5)
                 IF ABS(d2rnd - d2) < .000001 THEN
                  IF a + b + c + d + d1 + d2 < 24 THEN
                   PRINT a; b; c; d, d1; d2, a + b + c + d + d1 + d2
                   PRINT USING "####.####"; cvdg(angle1(d1, a, b)); cvdg(angleA + angleD);
                   PRINT USING "####.####"; cvdg(angle1(d1, c, d)); cvdg(angleB + angleC)
                   PRINT
                  END IF
                 END IF
              END IF
            NEXT
         NEXT t2
    NEXT d1
  NEXT a
NEXT t

FUNCTION angle1 (a, b, c)
   cosA = (b * b + c * c - a * a) / (2 * b * c)
   IF cosA = 0 THEN angleA = pi / 2: EXIT FUNCTION
   sinA = SQR(1 - cosA * cosA)
   angleA = ATN(sinA / cosA)
   IF angleA < 0 THEN angleA = angleA + pi
   angle1 = angleA
END FUNCTION

FUNCTION cvdg (x)
   cvdg = x * 180 / pi
END FUNCTION

 

  sides     diagonals     total
 
2  3  2  4    4  4          19
104.4775 104.4775  75.5225  75.5225   angles
2  4  4  5    3  2          20
 46.5675  28.9550  36.8699 157.6076   angles
2  4  2  3    4  4          19
 75.5225  75.5225 104.4775 104.4775   angles
2  4  3  5    4  2          20
 75.5225  28.9550  53.1301 112.3924   angles
3  4  2  1    2  4          16
 28.9550  75.5225  75.5225 180.0000   angles
 
 


The smallest without a 180 angle would have sides 2, 3, 2, 4, giving a weighted total of 19. Then, 2, 4, 4, 5 or 2, 4, 3, 5 would have a total of 20.


  Posted by Charlie on 2013-07-25 15:57:08
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