Question

point j lies on $overline{hi}$ such that $hj:ij$ is $6:1$. graph j.

Explanation:

1. Assume the coordinates of point \(H=(x_1,y_1)\) and point \(I=(x_2,y_2)\). First, find the coordinates of the point - dividing the line - segment \(\overline{HI}\) in the ratio \(m:n\) (here \(m = 6\) and \(n = 1\)) using the section - formula:

• The section formula for a point \(J=(x,y)\) that divides the line - segment joining \(H=(x_1,y_1)\) and \(I=(x_2,y_2)\) in the ratio \(m:n\) is given by \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\).
• Let's assume \(H=(2,2)\) and \(I=(18,14)\) (by observing the grid).
• Calculate the \(x\) - coordinate of \(J\):
• ## Step1: Substitute values into \(x\) - formula
• \(x=\frac{6\times18 + 1\times2}{6 + 1}=\frac{108+2}{7}=\frac{110}{7}\approx15.71\).
• Calculate the \(y\) - coordinate of \(J\):
• ## Step2: Substitute values into \(y\) - formula
• \(y=\frac{6\times14+1\times2}{6 + 1}=\frac{84 + 2}{7}=\frac{86}{7}\approx12.29\).

1. Locate the point \(J\) on the graph:

• On the grid, we can estimate the position of the point \(J\) based on the calculated coordinates. The point \(J\) is approximately at \((16,12)\) (rounding the calculated values to the nearest whole - number for graphing on the grid).

Answer:

Locate the point on the line - segment \(\overline{HI}\) approximately at \((16,12)\) on the given grid.