Answer:
The lengths are
8 + 8[root]3 , 12 + 8[root]3 and 20 + 8[root]3.
Step-by-step explanation:
Let the side against the barn be xft
We were told that the adjacent side is 4ft more than the side against the barn. This means it has a length of (x + 4)ft.
Lastly, we are told that the hypotenuse is 8ft longer than the side along the barn I.e (x + 4 + 8)ft = ( x + 12)ft.
Since the three lengths form a triangle, the square of the hypotenuse equals the square of the adjacent side plus the square of the opposite side. This is the pythagoras' theorem.
Mathematically, this can be written as follows:
(x + 12)^2 = x^2 + (x + 4)^2
This yields x^2 + 24x + 144 = x^2 + x^2 + 8x + 16
x^2 + 24x + 144 = 2x^2 + 8x + 16
Rearranging this will yield:
x^2 - 16x - 128 = 0
Using the quadratic formula yields the following values for x :
8 - 8[root]3 and 8 + 8[root 3]
Since the first is negative, we discard it and the only correct value of x is 8 + 8[root]3
Now the other length is 12 + 8[root]3 and 20 + 8[root]3.
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