In the figure shown, all line segments meet at right angles, and the area of each rectangular region is one-half the area of the next larger region. What fractional part of square region ABCD is shaded?
The fractional part of square region ABCD that is shaded is 17/64.
The problem states that each rectangular region's area is half that of the next larger region. This geometric progression allows us to calculate the shaded area relative to the total area of square region ABCD.
This choice suggests that a larger portion of square ABCD is shaded than what the geometric relationship permits. Since the areas decrease by half, a fraction of 5/16 does not align with the calculated areas of the regions, which are smaller than this value.
While this option represents a fraction less than 1/2, it still exceeds the expected shaded area based on the given relationship of halving areas. The calculations for the shaded area, derived from the total area of square ABCD, yield a value smaller than 9/32.
This choice correctly represents the fractional part of square region ABCD that is shaded. As each subsequent rectangle's area is half of the previous one, the total shaded area accumulates to 17/64 of the whole square area, aligning perfectly with the relationships defined in the problem.
This option indicates an even smaller portion of the square area being shaded than what is calculated. Given the relationships established in the geometric progression, 33/128 does not accurately reflect the total shaded area derived from the larger rectangular areas.
The problem illustrates how the areas of successively smaller rectangles relate to the overall area of square ABCD. By understanding the halving relationship among the areas, we determine that the total shaded area is accurately represented as 17/64. This conclusion maintains consistency with the geometric constraints laid out in the question, ensuring the solution reflects the correct fractional part shaded within the larger square.
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