In the 1142nd exercise (1. in this number on page 201), in the kite, we draw lines parallel to the diagonal $AC$ at a distance $x$ from it on both sides. These two parallel lines determine a quadrilateral $KLMN$ inscribed in the kite $ABCD$. How does the area of this quadrilateral change if the two parallel lines move away from $AC$ in equal measure?
The triangles $A B C$ and $A D C$ are symmetric with respect to $A C$, and $K$ and $L$, as well as $M$ and $N$, are pairs of symmetric points such that $K L=M N$; therefore, the quadrilateral $K L M N$ is a rectangle, with sides: $K L=M N=2 x$ and $K M=L N=z$.

![](https://cdn.mathpix.com/cropped/2024_05_02_26ca8659505e59ec33d9g-1.jpg?height=572&width=393&top_left_y=294&top_left_x=844)

Now

$$
\begin{aligned}
& A B C \Delta \sim K B M \Delta \text { and therefore } \\
& K M: A C=(B O-x): B O \\
& z: 10=(3-x): 3
\end{aligned}
$$

so

$$
z=\frac{10}{3}(3-x)
$$

The area of the rectangle is:

$$
y=2 x z=\frac{20 x}{3}(3-x)=-\frac{20}{3} x^{2}+20 x
$$

We need to examine the variation of this function when $0 < x < 3$.

| $x$ | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |
| --- | --- | --- | --- | --- | --- | --- |
| $y$ | 8.33 | 13.33 | 15 | 13.33 | 8.33 | 0 |

The maximum value of $y$ is 15, which occurs when $x = 1.5$.

| $x$ | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |
| --- | --- | --- | --- | --- | --- | --- |
| $\max$. | 14.4 | 12.6 | 9.6 | 5.4 | 0 |

![](https://cdn.mathpix.com/cropped/2024_05_02_26ca8659505e59ec33d9g-1.jpg?height=488&width=410&top_left_y=1900&top_left_x=844)