1. Let's start by understanding the problem: We have a line segment between points $P_1(-5, -4)$ and $P_2(3, 4)$ divided into four equal parts. We want to find the point of division nearest to $P_2$. 2. To find points dividing a segment in a ratio, we use the section formula: If a point $P$ divides the segment $P_1P_2$ in the ratio $r_1:r_2$, then $$x = \frac{r_2 x_1 + r_1 x_2}{r_1 + r_2}, \quad y = \frac{r_2 y_1 + r_1 y_2}{r_1 + r_2}$$ 3. Here, $r_1$ and $r_2$ represent how the segment is split. Specifically, $r_1$ is the part of the segment from $P$ to $P_2$, and $r_2$ is the part from $P_1$ to $P$. 4. Since the segment is divided into 4 equal parts, the points divide the segment in ratios like $1:3$, $2:2$, $3:1$ depending on which division point you consider. 5. To find the point nearest to $P_2$, we look at the division point that is one part away from $P_2$, so the ratio is $r_1:r_2 = 3:1$ (meaning $P$ is 3 parts from $P_1$ and 1 part from $P_2$). 6. So, $r_1$ is the number of parts from $P$ to $P_2$, and $r_2$ is the number of parts from $P_1$ to $P$. This is the easy way to identify $r_1$ and $r_2$: count how many equal parts lie on each side of the division point. 7. Using the formula with $r_1=3$ and $r_2=1$ gives the coordinates of the point nearest to $P_2$ as $$x = \frac{1 \times (-5) + 3 \times 3}{3 + 1} = 1, \quad y = \frac{1 \times (-4) + 3 \times 4}{3 + 1} = 2$$ This method helps you identify $r_1$ and $r_2$ clearly and apply the section formula easily.