Question

point f lies on (overline{de}) such that (df:ef) is (1:2). graph f.

Explanation:

Step1: Assume coordinates of D and E

Let \(D=(3,18)\) and \(E=(16,6)\) (by observing the grid - points).

Step2: Use the section formula

The section formula for a point \(F=(x,y)\) that divides the line - segment joining \(D=(x_1,y_1)\) and \(E=(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here \(m = 1\) and \(n = 2\), \(x_1=3,y_1 = 18,x_2=16,y_2 = 6\).
\[x=\frac{1\times16+2\times3}{1 + 2}=\frac{16 + 6}{3}=\frac{22}{3}\approx7.33\]
\[y=\frac{1\times6+2\times18}{1 + 2}=\frac{6+36}{3}=\frac{42}{3}=14\]

Answer:

The coordinates of point \(F\) are approximately \((7.33,14)\) and it can be graphed on the grid at this location.