The longer leg of a right triangle is 4cm longer than the shorter leg. the hypotenuse is 8cm longer than the shorter leg. find the side lengths of the triangle

Longer leg = x + 4

Shorter leg = x

Hypotenuse = x + 8

Using Pythagorean Theorem:

a^2 + b^2 = c^2

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

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

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

2x^2 + 8x = x^2 + 16x + 48

2x^2 - 8x = x^2 + 48

x^2 - 8x = 48

x^2 - 8x - 48 = 0

You can complete the square from here or use the quadratic formula.

Completing the square:

x^2 - 8x = 48

x^2 - 8x + (-8/2) ^2 = 48 + (-8/2) ^2

x^2 - 8x + 16 = 48 + 16

(x - 4) (x - 4) = 64 or (x - 4) ^2 = 64

x - 4 = + √64 OR x - 4 = - √64

x - 4 = + 8 OR x - 4 = - 8

x = 12 OR x = - 4

However, you can't use negative 4 as a length because your length needs to be a positive.

So x will be 12.

Shorter leg: 12

Longer leg: 12 + 4

Hypotenuse: 12 + 8