SOLUTION: What is the best approximation of the perimeter of a right isosceles triangle if its hypotenuse is 12 units long? what is its area?

Question 260707: What is the best approximation of the perimeter of a right isosceles triangle if its hypotenuse is 12 units long? what is its area?
Found 2 solutions by Fombitz, drk:
Answer by Fombitz(32388) (Show Source): You can put this solution on YOUR website!
For a right isoceles triangle,

or approximately

.
.
.

Answer by drk(1908) (Show Source): You can put this solution on YOUR website!
The issue here is that we don't know the other two sides. By the triangle inequality theorem, we know that they are greater than 12.
we also know a^2 + b^2 = c^2, where c = 12
We have several options
a = 1, b = sqrt(143), c = 12, P =
a = 2, b = sqrt(140), c = 12, P =
a = 3. b = sqrt(135), c = 12, P =
a = 4, b = sqrt(128), c = 12, P =
a = 5, b = sqrt(119), c = 12, P =
a = 6, b = sqrt(108), c = 12, P =
a = 7, b = sqrt(95), c = 12, P =
a = 8, b = sqrt(80), c = 12, P =
a = 9, b = sqrt(63), c = 12, P =
a = 10, b = sqrt(44), c = 12, P =
a = 11, b = sqrt(23), c = 12, P =
----
The average of all perimeters is ~ 26.64377
The missing sides a and b could be ~ 7.821
--
we use herons formula to estimate the area
we need S = (7.821 + 7.821 + 12)/2 = 13.821
A ~ sqrt(13.821(13.821-7.821)(13.821-7.821)(13.821-12))
A ~ 30.1089
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