Final answer:
The lengths of the sides of a right triangle with consecutive even integer sides are 6, 8, and 10. This is found by using the Pythagorean theorem and solving for the smallest even integer that satisfies the consecutive even integer condition.
Step-by-step explanation:
If the lengths of the sides of a right triangle are consecutive even integers, we can represent them as x, x+2, and x+4 where x is the smallest even integer. By the Pythagorean theorem, we know that the sum of the squares of the two shorter sides is equal to the square of the longest side, the hypotenuse. Therefore, we have x^2 + (x+2)^2 = (x+4)^2.
Solving for x, we first expand the equation: x^2 + x^2 + 4x + 4 = x^2 + 8x + 16. This simplifies to 2x^2 + 4x - 12 = 0. Dividing by 2, we get x^2 + 2x - 6 = 0. Factorizing the quadratic equation, we obtain (x+3)(x-1) = 0, which gives us two possible solutions for x: -3 and 1.
Since we are looking for even integers, we disregard -3 and take x=1, which isn't even. Hence, we adjust x to the next even number, which is 2. The three consecutive even integers are thus 2, 4, and 6, which satisfy the requirement of the question and the Pythagorean theorem. Therefore, the lengths of the sides of this right triangle are 6, 8, and 10.