Answer: Let's use the Pythagorean theorem to solve this problem.
Let x be the common factor of the ratio 3:4, so the sides of the rectangle are 3x and 4x.
The Pythagorean theorem states that for any right triangle, the sum of the squares of the two shorter sides is equal to the square of the length of the hypotenuse (the longest side).
So, for the rectangle with sides 3x and 4x, we have:
(3x)^2 + (4x)^2 = (diagonal)^2
9x^2 + 16x^2 = 100
25x^2 = 100
x^2 = 4
Taking the square root of both sides, we get:
x = 2
Therefore, the sides of the rectangle are:
3x = 3(2) = 6 cm
4x = 4(2) = 8 cm
So, the length and width of the rectangle are 6 cm and 8 cm, respectively.