To build the paper kite represented in the figure, Eduardo started by painting a rectangle ABCD on a sheet of paper. Then, he extended each side of the rectangle, tripling its length, and obtained the quadrilateral $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_4e05cbad380df980abcfg-7.jpg?height=452&width=782&top_left_y=542&top_left_x=474)

Figure 20.1

Knowing that the area of rectangle $A B C D$ is $200 \mathrm{~cm}^{2}$, what is the area of the kite constructed by Eduardo?
Observe that the triangles $A A^{\prime} D^{\prime}$ and $C C^{\prime} B^{\prime}$ are equal. Similarly, the triangles $B B^{\prime} A^{\prime}$ and $D D^{\prime} C^{\prime}$ are equal.

Thus, if $X$ and $Y$ are points such that $A^{\prime} B B^{\prime} X$ and $A^{\prime} A D^{\prime} Y$ are rectangles (Figure 20.2), the area of the kite is equal to the sum of the areas of these rectangles plus the area of the rectangle $A B C D$, and each of these rectangles can be divided into $3 \times 2=6$ rectangles equal to $A B C D$.

We conclude that the area of the kite is $(6+6+1) \times 200=2600 \mathrm{~cm}^{2}$.

Suggestion: Show that the area of each of the four triangles is equal to three times the area of the rectangle $A B C D$.

Helpful Facts: By constructing a diagonal of a rectangle, it is divided into two triangles of the same area.

![](https://cdn.mathpix.com/cropped/2024_05_01_4e05cbad380df980abcfg-7.jpg?height=454&width=346&top_left_y=1007&top_left_x=1506)

Figure 20.2