Question

the longer leg of a right triangle is 4 ft longer than the shorter leg. the hypotenuse is 8 ft longer than the shorter leg. find the side lengths of the triangle.
length of the shorter leg: 12 ft
length of the longer leg: 16 ft
length of the hypotenuse: 20 ft

Explanation:

Step1: Define variables for sides

Let $x$ = shorter leg length (ft).
Longer leg: $x+4$, Hypotenuse: $x+8$

Step2: Apply Pythagorean theorem

$$x^2 + (x+4)^2 = (x+8)^2$$

Step3: Expand all terms

$$x^2 + x^2 + 8x + 16 = x^2 + 16x + 64$$

Step4: Simplify to quadratic equation

$$x^2 - 8x - 48 = 0$$

Step5: Factor the quadratic

$$(x-12)(x+4) = 0$$

Step6: Solve for valid $x$

Length cannot be negative, so $x=12$

Step7: Calculate other sides

Longer leg: $12+4=16$, Hypotenuse: $12+8=20$

Answer:

Length of the shorter leg: 12 ft
Length of the longer leg: 16 ft
Length of the hypotenuse: 20 ft