a) In the figure below, there are three squares with sides 9, 6, and $x$. Determine the value of $x$.

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-08.jpg?height=523&width=920&top_left_y=1925&top_left_x=441)
b) Marcelo continues the drawing above and draws more squares (many!). Since these became too small, they cannot be seen, but we show some in the figure below. What is the length of the side of the $2014^{\circ}$ square, counting from left to right?

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-09.jpg?height=526&width=914&top_left_y=488&top_left_x=648)

#
Solution

a) Observe the figure:

![](https://cdn.mathpix.com/cropped/2024_05_01_463eeeabc25d438954b1g-33.jpg?height=534&width=920&top_left_y=407&top_left_x=441)

By similarity of triangles, we have

$$
\frac{6-x}{x}=\frac{3}{6}
$$

Therefore, $x=4$.

b) We have that the squares are similar in a ratio of $\frac{2}{3}$. That is, each new square has a side length equal to $\frac{2}{3}$ of the previous side. Thus, the $2014^{\varrho}$ square will have a side equal to

$$
9\left(\frac{2}{3}\right)^{2013}
$$