Final answer:
To find the length of the segment joining the legs of the trapezoid, we use the fact that the ratio of the areas of the two regions is equal to the ratio of the lengths of their corresponding bases. By solving an equation and calculating the value, we find that the length of the segment is 75 units. Finally, we calculate (75^2)/100 to find the greatest integer that does not exceed x^2/100, which is 56.
Step-by-step explanation:
To find the length of the segment joining the legs of the trapezoid that divides the trapezoid into two regions of equal area, we can use the concept that the ratio of the areas of the two regions is equal to the ratio of the lengths of their corresponding bases.
Let one base of the trapezoid be x units, then the other base is x + 100 units. The length of the segment joining the midpoints of the legs is equal to the average of the two bases, so it is (x + x + 100)/2 = (2x + 100)/2 = x + 50 units.
Since the areas of the two regions are in the ratio 2:3, we can set up the equation: 2x/(x + 50) = 2/3. Cross multiplying gives us 3(2x) = 2(x + 50), which simplifies to 6x = 2x + 100. Solving for x, we find that x = 25.
Therefore, the length of the segment joining the legs of the trapezoid that is parallel to the bases and divides the trapezoid into two regions of equal area is x + 50 = 25 + 50 = 75 units. Finally, we calculate (75^2)/100 = 5625/100 = 56.25, and the greatest integer that does not exceed 56.25 is 56.