The length of the longer leg of a right triangle is 13 ft more than three times the length of the shorter leg. The length of the hypotenuse is 14 ft more than three times the length of the shorter leg. Find the side lengths of the triangle.
If the shorter leg is x, then we have
x^2 + (3x+13)^2 = (3x+14)^2
x^2 + 9x^2 + 78x + 169 = 9x^2 + 84x + 196
x^2 - 6x - 27 = 0
Now finish it off
To solve this problem, 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 denote the length of the shorter leg as x.
According to the information given in the problem, the longer leg is 13 ft more than three times the length of the shorter leg. So, the length of the longer leg is 3x + 13 ft.
The hypotenuse is said to be 14 ft more than three times the length of the shorter leg. Therefore, the length of the hypotenuse is 3x + 14 ft.
Now, we can use the Pythagorean Theorem:
(3x + 13)^2 + x^2 = (3x + 14)^2
Simplifying this equation:
9x^2 + 78x + 169 + x^2 = 9x^2 + 84x + 196
Combining like terms and simplifying further:
10x^2 + 78x + 169 = 9x^2 + 84x + 196
Subtracting the common terms on both sides and simplifying:
x^2 - 6x - 27 = 0
Now, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values from our equation:
x = (6 ± √((-6)^2 - 4(1)(-27))) / (2(1))
Simplifying further:
x = (6 ± √(36 + 108)) / 2
x = (6 ± √144) / 2
x = (6 ± 12) / 2
Now we have two possible values for x: x = (6 + 12) / 2 = 9 or x = (6 - 12) / 2 = -3. However, the length cannot be negative, so we take x = 9 as the length of the shorter leg.
Using this value, we can find the lengths of the other sides:
Length of the longer leg = 3x + 13 = 3(9) + 13 = 40 ft
Length of the hypotenuse = 3x + 14 = 3(9) + 14 = 41 ft
So, the side lengths of the triangle are:
Shorter leg: 9 ft
Longer leg: 40 ft
Hypotenuse: 41 ft