2. 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 x = hypotenuse
let y = the length of the other leg
we have two equations, two unknowns here.
recall that perimeter of a triangle is just,
P = a + b + c
substituting,
38.7 = x + y + 12.5
x + y = 38.7 - 12.5
x + y = 26.2
x = 26.2 - y : : equation (1)
recall that for any right triangle, we can solve for hypotenuse using the Pythagorean theorem:
c^2 = a^2 + b^2
substituting,
y^2 = x^2 + 12.5^2
y^2 = x^2 + 156.25 : equation (2)
now, we substitute eqn (1) to eqn (2):
y^2 = (26.2 - y)^2 + 156.25
y^2 = 686.44 - 52.4y + y^2 + 156.25
y^2 - y^2 + 52.4y = 842.69
52.4y = 842.69
y = 16.08 ft (hypotenuse)
x = 26.2 - y = 10.12 ft (other leg)
hope this helps~ :)
P = Perimeter
a = First side = 12.5 ft
b = Second side
c = Hypotenuse
c = sqrt ( a ^ 2 + b ^ 2 )
c = sqrt ( 156.25 + b ^ 2 )
P = a + b + c = 38.7
12.5 + b + sqrt ( 156.25 + b ^ 2 ) = 38.7
b + sqrt ( 156.25 + b ^ 2 ) = 38.7 -12.5
b + sqrt ( 156.25 + b ^ 2 ) = 26.2
sqrt ( 156.25 + b ^ 2 ) = 26.2 - b Square both sides
156.25 + b ^ 2 = ( 26.2 - b ) ^ 2
156.25 + b ^ 2 = 26.2 ^ 2 - 2 * 26.2 * b + b ^ 2
156.25 + b ^ 2 = 686.44 - 52.4 b + b ^ 2
b ^ 2 + 52.4 b - b ^ 2 = 686.44 - 156.25
52.4 b = 530.19 Divide both sides with 52.4
b = 530.19 / 52.4
b = 10.118129771 ft
c = sqrt ( a ^ 2 + b ^ 2 )
c = sqrt ( 12.5 ^ 2 + 10.118129771
^ 2 )
c = sqrt ( 156.25 + 102.37655 )
c = sqrt ( 258.62655 )
c = 16.08187 ft
To solve this problem, we will use the Pythagorean theorem. According to the theorem, 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 right triangle x.
Using the Pythagorean theorem, we can write the equation:
x^2 + 12.5^2 = hypotenuse^2 ...(Equation 1)
We also know that the perimeter of a triangle is the sum of the lengths of all its sides. In this case, the perimeter is given as 38.7 ft, and we know that one side is 12.5 ft. So, the sum of the other two sides (x + hypotenuse) should be equal to 38.7 ft:
x + 12.5 + hypotenuse = 38.7 ...(Equation 2)
Now, we have a system of two equations (Equation 1 and Equation 2) with two unknowns (x and hypotenuse). We can solve this system of equations to find the values of x and hypotenuse.
First, let's rearrange Equation 2 to solve for hypotenuse:
hypotenuse = 38.7 - x - 12.5
Now, substitute this value of hypotenuse into Equation 1:
x^2 + 12.5^2 = (38.7 - x - 12.5)^2
Expanding the square on the right side, we get:
x^2 + 12.5^2 = (38.7 - x - 12.5)(38.7 - x - 12.5)
Simplifying the equation further, we have:
x^2 + 156.25 = (26.2 - x)^2
Expanding and simplifying the right side:
x^2 + 156.25 = 686.44 - 52.4x + x^2
Now, combine the like terms on both sides, and then we can solve for x:
156.25 = 686.44 - 52.4x + x^2 - x^2
156.25 = 686.44 - 52.4x
Rearranging and isolating the x term:
52.4x = 686.44 - 156.25
52.4x = 530.19
Now, divide both sides of the equation by 52.4:
x = 10.12
So, the length of the other unknown side is approximately 10.12 ft.
To find the hypotenuse, substitute the value of x back into Equation 2:
hypotenuse = 38.7 - x - 12.5
hypotenuse = 38.7 - 10.12 - 12.5
hypotenuse = 38.7 - 22.62
hypotenuse = 16.08
Therefore, the length of the hypotenuse is approximately 16.08 ft.