SOLUTION: The perimeter of a right triangle with side lengths that are integers, and having the same area as a rectangle with dimensions 36 cm by 45 cm, is in cm...?

Question 1132966: The perimeter of a right triangle with side lengths that are integers, and having the same area as a rectangle with dimensions 36 cm by 45 cm, is in cm...?
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
The perimeter of a right triangle with side lengths that are integers has the same area as a rectangle with dimensions 36 cm by 45 cm.
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Area = 36*45 = 1620
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Area of triangle = b*h/2 = 1620
b*h = 3240
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b^2 + h^2 = c^2, all integers
h = 3240/b
b^2 + (3240/b)^2 = c^2
b^4 + 10497600 = b^2c^2
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That's an ugly equation.
Try the commonly known right triangles with integers sides:
3-4-5
Area = 6
1620/6 = 270
3240/270 = 12 NG
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5-12-13
Area = 30
3240/30 = 108 NG
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8-15-17
Area = 60
3240/60 = 54 NG
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Find a right triangle with integer sides, and 3240/area is a perfect square.
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Here are some possibilities:
11: 60 :61
12: 35 :37
13: 84 :85
15: 112 :113
16: 63 :65
17: 144 :145
19: 180 :181
20: 21 :29
20: 99 :101
21: 220 :221
24: 143 :145
28: 45 :53
28: 195 :197
32: 255 :257
33: 56 :65
36: 77 :85
39: 80 :89
44: 117 :125
48: 55 :73
51: 140 :149
52: 165 :173
57: 176 :185
60: 91 :109
60: 221 :229
65: 72 :97
84: 187 :205
85: 132 :157
88: 105 :137
95: 168 :193
96: 247 :265
104: 153 :185
105: 208 :233
115: 252 :277
119: 120 :169
120: 209 :241
133: 156 :205
140: 171 :221
160: 231 :281
161: 240 :289
204: 253 :325
207: 224 :305
=========================
It's possible that there is no solution, or that there's a mistake in the entry of the problem.

Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.

The area of the rectangle is equal to 36*45 = 1620 cm^2. It is , 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 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 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. ANSWER

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

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