7. The largest square above has sides of length 8 and is divided into the two shaded rectangles and two smaller squares labeled I and II. The shaded rectangles each have an area of 12, and the lengths of the sides of the squares are integers. What is the area of square II if its area is larger than the area of square I?
Answer: C
The area of square II is 25.
Square II has an area of 25, which means its side length is 5. This satisfies the condition that its area is larger than that of square I and fits within the total area constraints established by the largest square.
A) 9
An area of 9 corresponds to a side length of 3. If square I has a side length of 3, its area would be smaller than that of square II, which must be larger. However, this would not allow square II to be larger than 3 while still satisfying the total area constraints.
B) 16
An area of 16 means a side length of 4 for square II. If square I were to have a smaller area, such as 12 for the rectangles, square II would only be marginally larger than square I. This does not meet the requirement that square II's area must be larger than square I, which is a necessary condition.
C) 25
An area of 25 corresponds to a side length of 5 for square II. This area is larger than that of square I, and when considering the total area of the largest square (64), the remaining space after accounting for the rectangles aligns perfectly, confirming that square II can indeed be larger than square I.
D) 36
An area of 36 would mean a side length of 6 for square II. While this area is larger than that of square I, it would exceed the total area available when accounting for the two rectangles (each with an area of 12) and square I, thus violating the area constraints of the largest square.
Conclusion
The area of square II is definitively 25 as it satisfies the conditions of being larger than square I while remaining within the total area constraints of the largest square. The other options, while potentially larger or smaller, fail to meet either the size requirement relative to square I or the overall area limitations imposed by the largest square.