Explanation:
We know that the length of the mid-segment of a trapezoid is equal to the average of the lengths of its parallel sides. So, we can set up an equation using this property:
Mid-segment length = (Length of the parallel sides sum) / 2
Substituting the given values, we get:
7x - 6 = (x^2 + 36) / 2
Multiplying both sides by 2, we get:
14x - 12 = x^2 + 36
Rearranging the terms, we get:
x^2 - 14x + 48 = 0
We can factor this quadratic equation as:
(x - 6)(x - 8) = 0
Therefore, x = 6 or x = 8.
However, we need to check if these values of x satisfy the condition that the length of one of the parallel sides is x^2.
When x = 6, one of the parallel sides is 6^2 = 36, which is already given in the problem statement. So, this solution is valid.
When x = 8, one of the parallel sides is 8^2 = 64, which does not match the given value of 36. So, this solution is not valid.
Therefore, the only solution that satisfies all the given conditions is x = 6.