the length of the longer leg of a right triangle is 13 ft more than 3 times the length of the shorter leg. the length of the hypotenuse is 14 ft more than 3 times the length of the shorter leg. Find the side lengths of the triangle
If the short leg is x, we have
x^2 + (3x+13)^2 = (3x+14)^2
x = 9
To find the side lengths of the right triangle, let's assume the length of the shorter leg is represented by "x".
According to the given information:
- The length of the longer leg is 13 ft more than 3 times the length of the shorter leg. We can express this as: longer leg = 3x + 13.
- The length of the hypotenuse is 14 ft more than 3 times the length of the shorter leg. We can express this as: hypotenuse = 3x + 14.
Now, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Using this theorem, we can write the equation:
(shorter leg)^2 + (longer leg)^2 = (hypotenuse)^2
Substituting the values we found earlier:
x^2 + (3x + 13)^2 = (3x + 14)^2
To solve this equation, let's simplify the equation step by step:
Expand the squares:
x^2 + (9x^2 + 78x + 169) = (9x^2 + 84x + 196)
Combine like terms:
10x^2 + 78x + 169 = 9x^2 + 84x + 196
Move all terms to one side of the equation:
10x^2 + 78x + 169 - 9x^2 - 84x - 196 = 0
Combine like terms:
x^2 - 6x - 27 = 0
Now, we can solve this quadratic equation. Factoring or using the quadratic formula, we find that the solutions are x ≈ -2.27 and x ≈ 4.27.
Since we are dealing with the length of a side, we can disregard the negative value. Therefore, the length of the shorter leg is approximately 4.27 ft.
Using this value, we can calculate the length of the longer leg and the hypotenuse:
Longer leg = 3(4.27) + 13 ≈ 25.81 ft
Hypotenuse = 3(4.27) + 14 ≈ 26.81 ft
Therefore, the side lengths of the right triangle are as follows:
Shorter leg ≈ 4.27 ft
Longer leg ≈ 25.81 ft
Hypotenuse ≈ 26.81 ft