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.

Answer 1

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