Respuesta :
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.
Pythagoras is the sum of the square of two sides is equal to the square of the longest side. Then the dimensions of the right triangle are 21.85, 25.85, and 33.85.
What is a right-angle triangle?
It is a type of triangle in which one angle is 90 degrees and it follows the Pythagoras theorem and we can use the trigonometry function.
A goat enclosure is in the shape of a right triangle.
One leg of the enclosure is built against the side of the barn.
The other leg is 4 feet more than the leg against the barn.
The hypotenuse is 8 feet more than the leg along the barn.
Let the length of the barn be x.
Then another leg will be x + 4.
And the hypotenuse will be x + 4 + 8 = x + 12
Then according to Pythagoras theorem, we have
[tex]\begin{aligned} (x + 12)^2 &= x^{2} + (x+4)^2\\\\x^2 + 24x + 144 &= x^2 + x^2 + 8x + 16\\\\x^2 -16x -128 &= 0 \\\\x &= 21.85, -5.85\end{aligned}[/tex]
Neglect -5.85.
Then the dimension of the right triangle will be
x = 21.85
x + 4 = 21.85 + 4 = 25.85
x + 12 = 21.85 + 12 = 33.85
More about the right-angle triangle link is given below.
https://brainly.com/question/3770177
