# 6. Variant 1.

The vertices of the triangle have coordinates $A(1 ; 3.5), B(13.5 ; 3.5), C(11 ; 16)$. Consider horizontal lines given by the equations $y=n$, where $n$ is an integer. Find the sum of the lengths of the segments cut off on these lines by the sides of the triangle.
Answer: 78.

Solution.

Draw the line $y=4$, and let it intersect the triangle $A B C$ at points $F$ and $G$. Construct a rectangle $F K M G$ such that $K M$ passes through point $C$ parallel to the $O x$ axis. Let $L_{1}, L_{2}, L_{3}$ be the sums of the lengths of the segments cut by the lines $y=n$ in the triangles $F K C$, $F C G$, and $C M G$, respectively. Due to symmetry (equal triangles $K F C = F C H$ and $C H G = C M G$), $L_{1} + L_{3} = L_{2}$. Find the length of $F G$ from the similarity of triangles $A B C$ and $F C G$. Since $\frac{D C}{H C} = \frac{A B}{F G}$, then $F G = 12$, so $L_{2} = L_{1} + L_{3} = 12 \cdot 13 / 2 = 78$.

![](https://cdn.mathpix.com/cropped/2024_05_06_fc73715326e960a104bag-5.jpg?height=911&width=805&top_left_y=1415&top_left_x=606)