5. A square with an area of $9 \mathrm{~cm}^{2}$ is divided into 9 equal squares as shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_06_03_43483f4d1877a66488a2g-1.jpg?height=166&width=165&top_left_y=2050&top_left_x=951)

a) Calculate the sum of the areas of all possible squares in the figure.

b) Calculate the sum of the areas of all possible rectangles in the figure that are not squares.

Each task is scored out of 10 points.

The use of a pocket calculator or any reference materials is not allowed.
5. a) In the figure, the following squares can be counted: $\square$
9 squares with an area of $1 \mathrm{~cm}^{2}$,
1 POINT

![](https://cdn.mathpix.com/cropped/2024_05_30_ba0648e72c3c0e2bc002g-3.jpg?height=129&width=126&top_left_y=438&top_left_x=308)
4 squares with an area of $4 \mathrm{~cm}^{2}$,
1 POINT

![](https://cdn.mathpix.com/cropped/2024_05_30_ba0648e72c3c0e2bc002g-3.jpg?height=186&width=168&top_left_y=632&top_left_x=287)

1 square with an area of $9 \mathrm{~cm}^{2}$.

1 POINT

The sum of the areas of all squares is $9 \cdot 1 + 4 \cdot 4 + 1 \cdot 9 = 34 \mathrm{~cm}^{2}$.

2 POINTS

b) The number of rectangles that are not squares is:
![](https://cdn.mathpix.com/cropped/2024_05_30_ba0648e72c3c0e2bc002g-3.jpg?height=626&width=1542&top_left_y=1020&top_left_x=271)

2 rectangles with an area of $6 \mathrm{~cm}^{2}$,

![](https://cdn.mathpix.com/cropped/2024_05_30_ba0648e72c3c0e2bc002g-3.jpg?height=163&width=122&top_left_y=1506&top_left_x=1024)
2 rectangles with an area of $6 \mathrm{~cm}^{2}$.

1 POINT

The sum of the areas of all rectangles that are not squares is $12 \cdot 2 + 6 \cdot 3 + 4 \cdot 6 = 66 \mathrm{~cm}^{2}$. 2 POINTS TOTAL 10 POINTS