Question 267085: The length of the hypotenuse of a right-angled triangle is 240 units. The perimeter of the given triangle is a perfect square. If the perimeter of the given triangle is greater than 550 units, then which of the following can be the length of a side of the given right-angled triangle?
Answer by CharlesG2(834)   (Show Source):
You can put this solution on YOUR website! The length of the hypotenuse of a right-angled triangle is 240 units. The perimeter of the given triangle is a perfect square. If the perimeter of the given triangle is greater than 550 units, then which of the following can be the length of a side of the given right-angled triangle?
a^2 + b^2 = c^2 = 240^2 = 57600
a + b + 240 > 550
a + b > 310
a > 310 - b
b > 310 - a
first perfect square after 550 is 24^2 which is 576
set a + b + 240 equal to 576
a + b = 336 (576-240=336)
a = 336 - b
b = 336 - a (use this to subsitute)
a^2 + (336 - a)^2 = 57600
a^2 + 112896 - 672a + a^2 = 57600 (336^2=112896, 2*336=672)
2a^2 - 672a + 55296 = 0
a^2 - 336a + 27648 = 0
27648 = 2 * 13824
13824 = 2 * 6912
6912 = 2 * 3456
3456 = 2 * 1728
1728 = 2 * 864
864 = 2 * 432
432 = 2 * 216
216 = 2 * 108
108 = 2 * 54
54 = 2 * 27
27 = 3 * 3 * 3
27648 = 2^10 * 3^3 = 1024 * 27
144 = 2^4 * 3^2
27648 = 2^4 * 3^2 * 2^6 * 3 = 16 * 9 * 64 * 3 = 144 * 192
144 + 192 = 336
(a - 192)(a - 144)=0
a = 144 or a = 192
now b = 336 - a
so b = 192 or b = 144
side a needs to be a minimum of 144, and side b needs to be a minimum of 192
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