A toy manufacture needs a piece of plastic in the shape of a right triangle with the longer leg 4 cm more than the shorter leg and the hypotenuse 8 cm more than the shorter leg how long should the sides of the triangle be

Another Pythagorean Theorem problem.

To find the lengths of the sides of the triangle, we can create a system of equations based on the given information.

Let's assume the length of the shorter leg of the right triangle is x cm.

According to the given information:
- The longer leg is 4 cm more than the shorter leg, so its length would be x + 4 cm.
- The hypotenuse is 8 cm more than the shorter leg, so its length would be x + 8 cm.

Using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, we can set up an equation:

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

Now, let's solve this equation to find the value of x, which represents the length of the shorter leg of the triangle:

x^2 + (x^2 + 8x + 16) = (x^2 + 16x + 64)
2x^2 + 8x + 16 = x^2 + 16x + 64
2x^2 - x^2 + 8x - 16x + 16 - 64 = 0
x^2 - 8x - 48 = 0

Now we have a quadratic equation. We can solve it by factoring or using the quadratic formula. Factoring might be a bit tricky for this equation, so let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For this equation, a = 1, b = -8, and c = -48. Plugging these values into the quadratic formula:

x = (-(-8) ± √((-8)^2 - 4(1)(-48))) / (2(1))
x = (8 ± √(64 + 192)) / 2
x = (8 ± √256) / 2
x = (8 ± 16) / 2

Now, we have two values for x: (8 + 16) / 2 and (8 - 16) / 2. Simplifying these expressions:

x₁ = 24 / 2 = 12
x₂ = -8 / 2 = -4

Since we are dealing with lengths, a negative value doesn't make sense in this context. So, we discard x₂ = -4.

Therefore, the length of the shorter leg (x) is 12 cm. To find the lengths of the longer leg and the hypotenuse, we can substitute this value into the given conditions:

Longer leg = x + 4 = 12 + 4 = 16 cm
Hypotenuse = x + 8 = 12 + 8 = 20 cm

So, the sides of the right triangle should be 12 cm, 16 cm, and 20 cm.