We have a T-shaped figure composed of 22 squares with a side length of $1 \mathrm{~cm}$. A line $p$ passes through point $V$, dividing the figure into two parts, white and gray. Determine in what ratio the line $p$ divides the segment $A B$, given that the areas of the white and gray regions are equal.

(L. Šimünek)

![](https://cdn.mathpix.com/cropped/2024_04_17_89c7e734c425fc1afd89g-1.jpg?height=591&width=593&top_left_y=1943&top_left_x=800)
Triangles $V F G$ and $V I H$ are congruent (by the SAS theorem: $G V F$ and $H V I$ are vertex angles, $\left.|V F|=|V I|,|\nless V F G|=|\nless V I H|=90^{\circ}\right)$.

![](https://cdn.mathpix.com/cropped/2024_04_17_89c7e734c425fc1afd89g-2.jpg?height=591&width=577&top_left_y=384&top_left_x=814)

We can therefore mentally color triangle $V I H$ gray and triangle $V F G$ white without changing the relative representation of the colors.

After this adjustment, we have ten unit squares that are white, ten that are gray, and two that are bicolored. For both colors to occupy areas of the same area, segment $C G$ must divide rectangle $A B K F$ into two congruent trapezoids. Therefore,

$$
|D E|=|J E|=0.5 \text{. }
$$

Segment $F G$ is the midline of triangle $D E V$, so

$$
|F G|=\frac{|D E|}{2}=\frac{0.5}{2}=0.25.
$$

From the congruence of the mentioned trapezoids, it follows that $|B C|=|F G|=0.25$. It is clear that $|A C|=$ $=|A B|-|B C|=1-0.25=0.75$. Line $p$ divides segment $A B$ in the ratio

$$
\frac{0.75}{0.25}=\frac{3}{1}
$$

SOLUTION USING SIMILARITY: Triangles $V A C$ and $V I H$ are similar in the ratio 3:1. Let the area of triangle $V I H$ be $S$, then the area of triangle $V A C$ is $9S$. The areas of the gray and white regions are equal, so $10-S+9S=12-9S+S$, and thus

$$
S=\frac{1}{8}
$$

From the area formula for triangle $V I H$,

$$
S=\frac{1}{2}|V I| \cdot|I H|
$$

we get $|I H|=0.25$. From the similarity ratio of the mentioned triangles, it follows that $|A C|=$ $=3|H I|=3 \cdot 0.25=0.75$, so $|A C|=0.75$. Line $p$ divides segment $A B$ in the ratio

$$
\frac{0.75}{0.25}=\frac{3}{1}
$$