1. The problem is to understand the meaning of \(\tan \alpha\) based on the given graph which shows an angle \(\alpha\) between a line and the horizontal axis.\n\n2. By definition, \(\tan \alpha = \frac{\text{opposite side}}{\text{adjacent side}}\) in a right triangle. The tangent of angle \(\alpha\) is the ratio of the length of the side opposite \(\alpha\) to the length of the side adjacent to \(\alpha\).\n\n3. In the context of the graph, the line creates angle \(\alpha\) with the horizontal axis. The tangent line segment length shown corresponds to \(\tan \alpha\), which measures the vertical change divided by the horizontal distance (the slope) from the point where the angle is measured.\n\n4. Therefore, \(\tan \alpha\) represents the slope of the line, or how steep it is, calculated as the ratio of the vertical side to the horizontal side adjacent to the angle \(\alpha\).\n\nFinal answer: \(\tan \alpha\) is the ratio of the vertical side length to the horizontal side length adjacent to angle \(\alpha\), representing the slope of the line making angle \(\alpha\) with the horizontal axis.