Question

find the coordinates of point h on $overline{gi}$ such that the ratio of $overline{gh}$ to $overline{hi}$ is 1:4. g(4,6) i(1,2) show your work here

Explanation:

Step1: Recall section - formula

The section - formula for a point \(H(x,y)\) that divides the line - segment joining \(G(x_1,y_1)\) and \(I(x_2,y_2)\) 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, \(G(4,6)\), \(I(1,2)\), \(m = 1\), and \(n = 4\).

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

Substitute \(x_1 = 4\), \(x_2=1\), \(m = 1\), and \(n = 4\) into the \(x\) - coordinate formula:
\[x=\frac{1\times1+4\times4}{1 + 4}=\frac{1 + 16}{5}=\frac{17}{5}=3.4\]

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

Substitute \(y_1 = 6\), \(y_2 = 2\), \(m = 1\), and \(n = 4\) into the \(y\) - coordinate formula:
\[y=\frac{1\times2+4\times6}{1 + 4}=\frac{2+24}{5}=\frac{26}{5}=5.2\]

Answer:

The coordinates of point \(H\) are \((3.4,5.2)\)