the hypotenuse of a triangle is 24 ft. long. the lenght of one leg is 6ft. more tha the other. Find the lengths of the legs.
use the a^2 + b^2 = c^2
c being the length of the hypotenuse
To find the lengths of the legs of the triangle, we need to apply 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 legs.
Let's assume the length of one leg is x. According to the problem statement, the other leg's length is 6 ft more than x. So, the length of the other leg can be expressed as (x + 6).
Using the Pythagorean Theorem, we can write the equation as:
x^2 + (x + 6)^2 = 24^2
Simplifying the equation:
x^2 + (x^2 + 12x + 36) = 576
Combining like terms:
2x^2 + 12x + 36 = 576
Rearranging the equation:
2x^2 + 12x + 36 - 576 = 0
2x^2 + 12x - 540 = 0
Dividing the equation by 2:
x^2 + 6x - 270 = 0
Now, we can solve this quadratic equation to find the value of x. You can use the quadratic formula or factoring if possible. In this case, factoring might be a bit difficult due to the larger numbers involved, so using the quadratic formula is recommended:
x = (-b ± √(b^2 - 4ac)) / 2a
With a = 1, b = 6, and c = -270, let's substitute these values:
x = (-6 ± √(6^2 - 4*1*(-270))) / (2*1)
Simplifying:
x = (-6 ± √(36 + 1080)) / 2
x = (-6 ± √(1116)) / 2
x ≈ (-6 ± 33.42) / 2
Calculating both possible values:
x ≈ (-6 + 33.42) / 2 ≈ 13.71
x ≈ (-6 - 33.42) / 2 ≈ -19.71
Since lengths can't be negative, we ignore the negative value. Therefore, the length of one leg is approximately 13.71 ft.
Using this value, we can find the length of the other leg:
Length of the other leg = x + 6 ≈ 13.71 + 6 ≈ 19.71 ft.
Therefore, the lengths of the legs of the triangle are approximately 13.71 ft and 19.71 ft.