Question 1132966
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<pre>
The area of the rectangle is equal to 36*45 = 1620 cm^2.


It is {{{(1/2)*a*b}}},  where "a" and "b" are the legs of a triangle;

hence, a*b = 3240 cm^2.


Thus our task is to find right-angled triangles with integer sides "a" and "b" such that ab = 3240 and 
the hypotenuse  {{{sqrt(a^2 + b^2)}}}  is an integer number, too. 


The reasonable way to organize the search is to try all factors "a" and "b" of the number 3240 and check 
for each pair (a,b) whether the hypotenuse  {{{sqrt(a^2+b^2)}}} is integer.


3240 = 324*10 = 18^2*10 = 2^3*3^4*5.


It is easy to perform such search in Excel : see the Table below.

a    b= 3240/a  sqrt(a^2+b^2)
-----------------------------

1	3240	3240.00015432
2	1620	1620.00123457
4	810	810.00987648
8	405	405.07900464
3	1080	1080.00416666
6	540	540.03333230
12	270	270.26653511
24	135	137.11673858
9	360	360.11248243
18	180	180.89776118
36	90	96.93296653
108	30	112.08925015
27	120	123.00000000     <<<---===
54	60	80.72174428
108	30	112.08925015
324	10	324.15428425
5	648	648.01928984
10	324	324.15428425
20	162	163.22989922
40	81	90.33825325
15	216	216.52020691
30	108	112.08925015
60	54	80.72174428
120	27	123.00000000     <<<---===
45	72	84.90583019
90	36	96.93296653
180	18	180.89776118
540	6	540.03333230
135	24	137.11673858
270	12	270.26653511
540	6	540.03333230
1620	2	1620.00123457
3240	1	3240.00015432


From the table, there is, actually, only ONE such a triangle with the legs 27 and 120 centimeters and the hypotenuse of 123 centimeters.


Its perimeter is  27 + 120 + 127 = 270 centimeters.      <U>ANSWER</U>
</pre>

Solved.  &nbsp;&nbsp;&nbsp;&nbsp;// &nbsp;&nbsp;&nbsp;&nbsp;The primitive Pythagorean triple for this triangle is  &nbsp;&nbsp;(9, 40, 41).