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Triangles And Rectangles Pair (Posted on 2013-11-01) Difficulty: 2 of 5
A triangle and rectangle have the same area and perimeter. All sides are integers.

Find such a pair with the smallest area? And the next smallest?
Can you find such a pair with a right triangle?

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution computer solution Comment 1 of 1

The program uses Heron's formula for the area of the triangle:

DEFDBL A-Z
CLS
FOR peri = 4 TO 9999 STEP 2
   FOR w = 1 TO peri / 4
     l = peri / 2 - w
     area = l * w
     FOR s1 = 1 TO peri / 3
       FOR s2 = s1 TO (peri - s1) / 2
         s3 = peri - s1 - s2
         IF s1 + s2 > s3 THEN
          s = peri / 2
          tarea = SQR(s * (s - s1) * (s - s2) * (s - s3))
          IF tarea = area THEN
            PRINT l; w, s1; s2; s3; TAB(30); area;
            IF s1 * s1 + s2 * s2 = s3 * s3 THEN PRINT " right":  ELSE PRINT
          END IF
         END IF
       NEXT s2
     NEXT s1
   NEXT w
NEXT peri

finds as the first few, in order of increasing perimeter:

rect.:         triangle:     area
l   w            sides
6  2          5  5  6        12
12  4         10  10  12     48
18  6         15  15  18     108
21  6         13  20  21     126
24  8         20  20  24     192
30  10        25  25  30     300
36  12        30  30  36     432
42  12        26  40  42     504
42  14        35  35  42     588
52  12        25  51  52     624
48  16        40  40  48     768
54  18        45  45  54     972
60  20        50  50  60     1200
63  18        39  60  63     1134
60  21        53  53  56     1260
66  22        55  55  66     1452
72  24        60  60  72     1728
78  26        65  65  78     2028
84  24        52  80  84     2016
84  28        70  70  84     2352
90  30        75  75  90     2700
105  20       41  104  105   2100
95  30        68  87  95     2850
104  24       50  102  104   2496
96  32        80  80  96     3072
105  30       65  100  105   3150
102  34       85  85  102    3468
108  36       90  90  108    3888
114  38       95  95  114    4332
120  40       100  100  120  4800
150  12       37  130  157   1800
126  36       78  120  126   4536
126  36       81  113  130   4536
120  42       106  106  112  5040
126  42       105  105  126  5292
132  44       110  110  132  5808
138  46       115  115  138  6348
147  42       91  140  147   6174
156  36       75  153  156   5616
144  48       120  120  144  6912

None were indicated to be right triangles.

The smallest area was 12 and the next smallest 48.


  Posted by Charlie on 2013-11-01 11:27:22
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