The lengths of the three sides of a right triangle are given by three consecutive even integers. Find the lengths of the tree sides.
Read about Pythagorean triples, especially the section about scaling.
http://www.mathsisfun.com/pythagorean_triples.html
To solve this problem, we can start by letting the first even integer be x. Since the three consecutive even integers are consecutive, the second even integer will be x + 2, and the third even integer will be x + 4.
Notice that if we square each of these quantities and add them together, we should get a perfect square according to the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Therefore, the equation we need to solve is:
x^2 + (x + 2)^2 = (x + 4)^2
Expanding and simplifying, we have:
x^2 + x^2 + 4x + 4 = x^2 + 8x + 16
Combining like terms, we get:
2x^2 + 4x + 4 = x^2 + 8x + 16
Subtracting x^2 and 8x from both sides, we have:
x^2 - 4x - 12 = 0
This is now a quadratic equation, which we can solve. Factoring the equation gives:
(x - 6)(x + 2) = 0
So, x = 6 or x = -2. Since we are discussing lengths, we can ignore the negative value.
Therefore, the first even integer, x, is 6. The second even integer is x + 2 = 6 + 2 = 8, and the third even integer is x + 4 = 6 + 4 = 10.
Thus, the lengths of the three sides of the right triangle are 6, 8, and 10.