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 Triangles And Rectangles Pair (Posted on 2013-11-01)
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 No Rating

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 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:     areal   w            sides6  2          5  5  6        1212  4         10  10  12     4818  6         15  15  18     10821  6         13  20  21     12624  8         20  20  24     19230  10        25  25  30     30036  12        30  30  36     43242  12        26  40  42     50442  14        35  35  42     58852  12        25  51  52     62448  16        40  40  48     76854  18        45  45  54     97260  20        50  50  60     120063  18        39  60  63     113460  21        53  53  56     126066  22        55  55  66     145272  24        60  60  72     172878  26        65  65  78     202884  24        52  80  84     201684  28        70  70  84     235290  30        75  75  90     2700105  20       41  104  105   210095  30        68  87  95     2850104  24       50  102  104   249696  32        80  80  96     3072105  30       65  100  105   3150102  34       85  85  102    3468108  36       90  90  108    3888114  38       95  95  114    4332120  40       100  100  120  4800150  12       37  130  157   1800126  36       78  120  126   4536126  36       81  113  130   4536120  42       106  106  112  5040126  42       105  105  126  5292132  44       110  110  132  5808138  46       115  115  138  6348147  42       91  140  147   6174156  36       75  153  156   5616144  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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