The hypotenuse of a right triangle is 24 feet long. The length of one leg is 20 feet more than the other
Although you haven't asked a question, I assume you want to find the length of the legs. Use Pythagorean theorem.
x^2 + (x+20)^2 = 24^2
To find the lengths of the legs of the right triangle, 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 lengths of the two legs.
Let's assume one leg of the right triangle is x feet long. According to the problem, the other leg is 20 feet longer than x, which means it is (x + 20) feet long.
Now we can write the Pythagorean theorem equation:
(x)^2 + (x + 20)^2 = (24)^2
Expanding and simplifying the equation:
x^2 + (x^2 + 40x + 400) = 576
Combining like terms:
2x^2 + 40x + 400 = 576
Rearranging the equation to set it equal to zero:
2x^2 + 40x + 400 - 576 = 0
2x^2 + 40x - 176 = 0
To solve the quadratic equation, we can either factor it or use the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values from our equation:
x = (-(40) ± √((40)^2 - 4(2)(-176))) / (2(2))
Simplifying:
x = (-40 ± √(1600 + 1408)) / 4
x = (-40 ± √(3008)) / 4
Extracting the square root:
x = (-40 ± √(16 * 188)) / 4
x = (-40 ± 4√(47)) / 4
Simplifying:
x = -10 ± √(47)
Now we have two possible values for x.
If we take the positive square root:
x = -10 + √(47)
This gives us one leg of the triangle.
To find the other leg, we can use the fact that it is 20 feet longer than x:
x + 20 = -10 + √(47) + 20
Simplifying:
x + 20 = 10 + √(47)
Therefore, the lengths of the legs of the right triangle are approximately -10 + √(47) feet and 10 + √(47) feet.