5. On the sides of a rectangle with length $8 \mathrm{~cm}$ and width $6 \mathrm{~cm}$, points are placed starting from one vertex, each $2 \mathrm{~cm}$ apart. How many different lines do these points determine?

NATIONAL MATHEMATICS COMPETITIONPrimošten, April 4-6, 2016.6th grade-elementary school
First method:

On the longer sides, three points are highlighted, and on the shorter sides, two points different from the vertices $A, E$, $H$ and $L$ are highlighted.

![](https://cdn.mathpix.com/cropped/2024_05_30_c165b3842bf96502feaeg-06.jpg?height=408&width=491&top_left_y=384&top_left_x=814)

With the labels as shown in the figure, we conclude step by step:

The point $A$ and some other points of the rectangle determine 8 lines.

The points $B, C$, and $D$ and some other points (excluding $A$) determine $3 \cdot 9=27$ lines.

The point $E$ and some other points (excluding $A, B, C$, and $D$) of the rectangle determine 7 lines. The points $F$ and $G$ and some other points (excluding $A, B, C, D$, and $E$) determine $2 \cdot 6=12$ lines. The point $H$ and some other points (excluding $A, B, C, D, E, F$, and $G$) determine 3 lines.

The points $K, J$, and $I$ and some other points (excluding $A, B, C, D, E, F, G$, and $H$) determine $3 \cdot 2=6$ lines.

These points determine $8+27+7+12+3+6=63$ lines.