1. The problem is to identify how to determine $r_1$ and $r_2$ in the section formula for dividing a line segment. 2. The section formula states that if a point $P$ divides the line segment $P_1P_2$ in the ratio $r_1:r_2$, then the coordinates of $P$ are given by: $$x = \frac{r_2 x_1 + r_1 x_2}{r_1 + r_2}, \quad y = \frac{r_2 y_1 + r_1 y_2}{r_1 + r_2}$$ 3. Here, $r_1$ and $r_2$ represent the parts into which the segment is divided. Specifically, $r_1$ is the distance from $P$ to $P_1$, and $r_2$ is the distance from $P$ to $P_2$. 4. To identify $r_1$ and $r_2$, you need to know how the point $P$ divides the segment: - If $P$ divides $P_1P_2$ internally, $r_1$ and $r_2$ are positive and represent the ratio of the lengths $PP_1$ to $PP_2$. - If $P$ divides externally, one of $r_1$ or $r_2$ is negative. 5. In the example given, the ratio is $r_1:r_2 = 3:1$, meaning $P$ is three parts from $P_1$ and one part from $P_2$. 6. So, to identify $r_1$ and $r_2$, you must know or be given the ratio in which $P$ divides the segment or the distances from $P$ to $P_1$ and $P_2$. 7. Once you have the ratio or distances, plug them into the formula to find the coordinates of $P$. Final answer: $r_1$ and $r_2$ are the parts of the segment into which point $P$ divides $P_1P_2$, representing distances from $P$ to $P_1$ and $P_2$ respectively, and must be known or given to use the section formula.