Question

gh has endpoints g(-6, 1) and h(5, 6). point i divides gh into two parts with lengths in a ratio of 4:1. what are the two possible locations of i? (2.2, 1.4) (-0.2, 1.4) (2.8, 5) (-5.5, -2.5) (0.5, 3.5) (-3.8, 2)

Explanation:

Step1: Recall the section - formula

The section - formula for a point \(I(x,y)\) that divides the line - segment joining \(G(x_1,y_1)\) and \(H(x_2,y_2)\) in the ratio \(m:n\) is given by \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(G(-6,1)\) i.e., \(x_1=-6,y_1 = 1\), \(H(5,6)\) i.e., \(x_2 = 5,y_2=6\), and \(m:n = 4:1\) or \(m = 4,n = 1\).

Step2: Calculate the x - coordinate of \(I\)

\[x=\frac{4\times5+1\times(-6)}{4 + 1}=\frac{20-6}{5}=\frac{14}{5}=2.8\]

Step3: Calculate the y - coordinate of \(I\)

\[y=\frac{4\times6+1\times1}{4 + 1}=\frac{24 + 1}{5}=\frac{25}{5}=5\]
Also, when we consider the ratio \(m:n=1:4\),

Step4: Calculate the x - coordinate of \(I\) for \(m = 1,n = 4\)

\[x=\frac{1\times5+4\times(-6)}{1 + 4}=\frac{5-24}{5}=\frac{-19}{5}=-3.8\]

Step5: Calculate the y - coordinate of \(I\) for \(m = 1,n = 4\)

\[y=\frac{1\times6+4\times1}{1 + 4}=\frac{6 + 4}{5}=\frac{10}{5}=2\]

Answer:

\((2.8,5),(-3.8,2)\)