Question

the shorter leg of a right triangle is 8 cm shorter than the longer leg. the hypotenuse is 8 cm longer than the longer leg. find the side lengths of the triangle.
length of the shorter leg: cm
length of the longer leg: cm
length of the hypotenuse: cm

Explanation:

Step1: Define variables for sides

Let the longer leg length be $x$ cm.
Shorter leg: $x - 8$ cm, Hypotenuse: $x + 8$ cm.

Step2: Apply Pythagorean theorem

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

Step3: Expand all squared terms

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

Step4: Simplify the equation

$$2x^2 - 16x + 64 = x^2 + 16x + 64$$
$$x^2 - 32x = 0$$
$$x(x - 32) = 0$$

Step5: Solve for valid $x$

$x=0$ is invalid, so $x=32$ cm.

Step6: Calculate other sides

Shorter leg: $32 - 8 = 24$ cm, Hypotenuse: $32 + 8 = 40$ cm.

Answer:

Length of the shorter leg: 24 cm
Length of the longer leg: 32 cm
Length of the hypotenuse: 40 cm