Question

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.
length of the shorter leg
ft
length of the longer leg:
ft
length of the hypotenuse:
ft

Explanation:

Step1: Define variables for sides

Let the shorter leg be $x$ ft.
Longer leg: $3x + 13$ ft
Hypotenuse: $3x + 14$ ft

Step2: Apply Pythagorean theorem

$$x^2 + (3x + 13)^2 = (3x + 14)^2$$

Step3: Expand all squared terms

$$x^2 + 9x^2 + 78x + 169 = 9x^2 + 84x + 196$$

Step4: Simplify the equation

$$x^2 - 6x - 27 = 0$$

Step5: Factor the quadratic equation

$$(x - 9)(x + 3) = 0$$

Step6: Solve for valid $x$

Length cannot be negative, so $x = 9$

Step7: Calculate other sides

Longer leg: $3(9) + 13 = 40$ ft
Hypotenuse: $3(9) + 14 = 41$ ft

Answer:

Length of the shorter leg: 9 ft
Length of the longer leg: 40 ft
Length of the hypotenuse: 41 ft