6. Variant 1.

On a grid paper, a right triangle with sides 6 and 10 is drawn (see figure). Find the total length of the horizontal grid lines inside this triangle.

![](https://cdn.mathpix.com/cropped/2024_05_06_45605a1e1b2292ab56c7g-06.jpg?height=892&width=615&top_left_y=733&top_left_x=706)
Answer: 27.

Solution. Extend the triangle to form a rectangle.

![](https://cdn.mathpix.com/cropped/2024_05_06_45605a1e1b2292ab56c7g-07.jpg?height=941&width=631&top_left_y=169&top_left_x=693)

It is divided into two identical triangles, so the total length of the horizontal lines in the rectangle is twice the desired total length. In the rectangle, there are 9 lines, each with a length of 6. The desired total length is $\frac{9 \cdot 6}{2}=27$.