1. **Problem Statement:** Find the coordinates of the point that divides the segment joining $(-2,0)$ and $(3,4)$ in the given ratios from left to right. 2. **Formula:** The point dividing a segment between points $A(x_1,y_1)$ and $B(x_2,y_2)$ in ratio $m:n$ is given by: $$\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$$ 3. **Given points:** $A(-2,0)$ and $B(3,4)$. 4. **Calculate for each ratio:** **a) Ratio 1:1** $$x = \frac{1\times3 + 1\times(-2)}{1+1} = \frac{3 - 2}{2} = \frac{1}{2} = 0.5$$ $$y = \frac{1\times4 + 1\times0}{1+1} = \frac{4}{2} = 2$$ Point: $(0.5, 2)$ **b) Ratio 1:2** $$x = \frac{1\times3 + 2\times(-2)}{1+2} = \frac{3 - 4}{3} = \frac{-1}{3} = -0.333$$ $$y = \frac{1\times4 + 2\times0}{1+2} = \frac{4}{3} = 1.333$$ Point: $(-0.333, 1.333)$ **c) Ratio 2:1** $$x = \frac{2\times3 + 1\times(-2)}{2+1} = \frac{6 - 2}{3} = \frac{4}{3} = 1.333$$ $$y = \frac{2\times4 + 1\times0}{2+1} = \frac{8}{3} = 2.667$$ Point: $(1.333, 2.667)$ **d) Ratio 4:1** $$x = \frac{4\times3 + 1\times(-2)}{4+1} = \frac{12 - 2}{5} = \frac{10}{5} = 2$$ $$y = \frac{4\times4 + 1\times0}{4+1} = \frac{16}{5} = 3.2$$ Point: $(2, 3.2)$ **e) Ratio 1:3** $$x = \frac{1\times3 + 3\times(-2)}{1+3} = \frac{3 - 6}{4} = \frac{-3}{4} = -0.75$$ $$y = \frac{1\times4 + 3\times0}{1+3} = \frac{4}{4} = 1$$ Point: $(-0.75, 1)$ **f) Ratio 2:3** $$x = \frac{2\times3 + 3\times(-2)}{2+3} = \frac{6 - 6}{5} = 0$$ $$y = \frac{2\times4 + 3\times0}{2+3} = \frac{8}{5} = 1.6$$ Point: $(0, 1.6)$