The longer leg of a right triangle is 4m longer than the shorter leg. The hypotenuse is 8m longer than the shorter leg. Find the side lengths of the triangle.
Let x be the length of the shorter leg.
The longer leg is then x + 4.
The hypotenuse is x + 8.
According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides, so we have:
x^2 + (x + 4)^2 = (x + 8)^2
Expanding this equation gives:
x^2 + x^2 + 8x + 16 = x^2 + 16x + 64
Simplifying further gives:
2x^2 + 8x + 16 = x^2 + 16x + 64
Rearranging terms gives:
x^2 - 8x - 48
This is a quadratic equation, which we can solve by factoring or using the quadratic formula. Factoring, we get:
(x - 12)(x + 4) = 0
Therefore:
x = 12 or x = -4
Since the length of a side cannot be negative, the only valid solution is x = 12.
Therefore, the lengths of the sides are:
Shorter leg: x = 12m
Longer leg: x + 4 = 16m
Hypotenuse: x + 8 = 20m