1. **Problem statement:** We have a line segment joining points $P_1(-5, -4)$ and $P_2(3, 4)$, divided into four equal parts. We need to find the point of division nearest to $P_2$. 2. **Formula used:** To divide a line segment into $n$ equal parts, the coordinates of the division points can be found using the section formula: $$ (x, y) = \left(x_1 + k \frac{x_2 - x_1}{n}, y_1 + k \frac{y_2 - y_1}{n}\right) $$ where $k = 1, 2, ..., n-1$. 3. **Apply the formula:** Here, $n=4$, $x_1 = -5$, $y_1 = -4$, $x_2 = 3$, $y_2 = 4$. Calculate the increments: $$ \frac{x_2 - x_1}{4} = \frac{3 - (-5)}{4} = \frac{8}{4} = 2 $$ $$ \frac{y_2 - y_1}{4} = \frac{4 - (-4)}{4} = \frac{8}{4} = 2 $$ 4. **Find the division points:** - For $k=1$: $$ (x, y) = (-5 + 1 \times 2, -4 + 1 \times 2) = (-3, -2) $$ - For $k=2$: $$ (x, y) = (-5 + 2 \times 2, -4 + 2 \times 2) = (-1, 0) $$ - For $k=3$: $$ (x, y) = (-5 + 3 \times 2, -4 + 3 \times 2) = (1, 2) $$ 5. **Nearest point to $P_2$:** The division point nearest to $P_2(3,4)$ is the one with $k=3$, which is $(1, 2)$. **Final answer:** The point of division nearest to $P_2$ is **$(1, 2)$**.