The shorter leg of a right triangle is 8 ft shorter than the longer leg. The hypotenuse is 8 ft longer than the longer leg. Find the side lengths of the triangle.
Length of the shorter leg:
Length of the longer leg:
Length of the hypotenuse:
Recall the Pythagorean Theorem. If the longer leg is x, then
(x-8)^2 + x^2 = (x+8)^2
Now just solve for x as usual.
Let's denote the length of the longer leg as x.
According to the problem, the shorter leg is 8 feet shorter than the longer leg, so its length would be x - 8 feet.
Similarly, the hypotenuse is 8 feet longer than the longer leg, so its length would be x + 8 feet.
Therefore, the lengths of the sides of the right triangle are as follows:
Length of the shorter leg: x - 8 feet
Length of the longer leg: x feet
Length of the hypotenuse: x + 8 feet
To find the side lengths of the right triangle, let's create variables to represent the lengths of the legs. Let's call the shorter leg "x", and the longer leg "y".
We know that the shorter leg is 8 feet shorter than the longer leg, so we can write the equation: x = y - 8.
We also know that the hypotenuse is 8 feet longer than the longer leg, so we can write the equation: hypotenuse = y + 8.
Using the Pythagorean theorem, we know that in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The Pythagorean theorem equation is: x^2 + y^2 = hypotenuse^2.
Now, let's substitute the values we have into the Pythagorean theorem equation: (y - 8)^2 + y^2 = (y + 8)^2.
Expanding and simplifying this equation gives us: y^2 - 16y + 64 + y^2 = y^2 + 16y + 64.
Combining like terms and simplifying further, we get: 2y^2 - 32y + 64 = y^2 + 16y + 64.
Moving all terms to one side gives us: 2y^2 - y^2 - 32y - 16y + 64 - 64 = 0.
Simplifying further, we have: y^2 - 48y = 0.
Factoring out a "y" gives us: y(y - 48) = 0.
Setting each factor equal to zero gives us two possible solutions: y = 0 or y - 48 = 0.
Since the length of a side of a triangle cannot be zero, we discard y = 0 as a solution.
So, we solve y - 48 = 0 to find the value of y: y = 48.
Now that we know the length of the longer leg, we can find the length of the shorter leg by substituting y = 48 into the equation x = y - 8: x = 48 - 8 = 40.
Therefore, the side lengths of the triangle are as follows:
Length of the shorter leg: 40 ft
Length of the longer leg: 48 ft
Length of the hypotenuse: 48 + 8 = 56 ft.