Question

point h lies on (overline{fg}) such that (fh:gh) is (2:1). graph h.

Explanation:

Step1: Assume coordinates

Let \(F=(x_1,y_1)=(3,12)\) and \(G=(x_2,y_2)=(15,6)\). Since \(FH:GH = 2:1\), then \(H\) divides the line - segment \(\overline{FG}\) in the ratio \(m:n = 2:1\).

Step2: Use section - formula for \(x\) - coordinate

The formula for the \(x\) - coordinate of a point \(H\) that divides the line - segment joining \((x_1,y_1)\) and \((x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\). Substitute \(m = 2\), \(n = 1\), \(x_1=3\), and \(x_2 = 15\):
\[x=\frac{2\times15+1\times3}{2 + 1}=\frac{30 + 3}{3}=\frac{33}{3}=11\]

Step3: Use section - formula for \(y\) - coordinate

The formula for the \(y\) - coordinate of a point \(H\) that divides the line - segment joining \((x_1,y_1)\) and \((x_2,y_2)\) in the ratio \(m:n\) is \(y=\frac{my_2+ny_1}{m + n}\). Substitute \(m = 2\), \(n = 1\), \(y_1=12\), and \(y_2 = 6\):
\[y=\frac{2\times6+1\times12}{2 + 1}=\frac{12 + 12}{3}=\frac{24}{3}=8\]

Answer:

The coordinates of point \(H\) are \((11,8)\). Graph the point \((11,8)\) on the given coordinate - plane.