one side of a right triangle is 12.5 ft the perimeter is 38.7 ft. what is the length of the hypotenuse and the other unknown side?
Let a,b, be the legs, and c the hypotenuse
a+b+c = 38.7
a + b + √(12.52+b2) = 38.7
√(12.52+b2) = 26.2 - b
12.52+b2 = b2 - 52.4b + 686.44
52.4b = 530.19
b = 10.1
a = 12.5
c = 16.1
To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's call the unknown side of the triangle "x". So we have one side as 12.5 ft, the other side as x ft, and the hypotenuse as a to be determined length.
According to the Pythagorean theorem, we can write the following equation:
12.5^2 + x^2 = a^2
We also know that the perimeter of the triangle is 38.7 ft, which means the sum of all three sides.
Perimeter = 12.5 + x + a
Combining these two equations, we have:
38.7 = 12.5 + x + a
Now, we need to isolate the variable "a" to determine the length of the hypotenuse.
To do this, let's subtract 12.5 and x from both sides of the equation:
38.7 - 12.5 - x = a
Simplifying this equation further:
26.2 - x = a
Now that we have the value for "a", we can substitute it back into the Pythagorean equation to solve for x:
12.5^2 + x^2 = (26.2 - x)^2
125 + x^2 = 686.44 - 52.4x + x^2
Combining like terms, we get:
0 = 686.44 - 52.4x - 125
Simplifying further:
0 = 561.44 - 52.4x
Now, we can solve for x:
52.4x = 561.44
Dividing both sides by 52.4:
x = 10.71 ft (rounded to two decimal places)
To find the length of the hypotenuse, we can substitute the value of x into the equation we found earlier:
a = 26.2 - x
a = 26.2 - 10.71
a = 15.49 ft (rounded to two decimal places)
Therefore, the length of the hypotenuse is approximately 15.49 ft and the length of the other unknown side is approximately 10.71 ft.