Question

what is the location of point g, which partitions the directed line segment from d to f into a 5:4 ratio? -1 0 2 3

Explanation:

Step1: Determine the coordinates of \(D\) and \(F\)

From the number - line, \(D=-2\) and \(F = 7\)

Step2: Use the section formula for a one - dimensional line segment

The formula for a point \(G\) that divides the line segment from \(x_1\) (coordinate of \(D\)) to \(x_2\) (coordinate of \(F\)) in the ratio \(m:n\) is \(G=\frac{mx_2+nx_1}{m + n}\)
Here, \(m = 5\), \(n=4\), \(x_1=-2\), \(x_2 = 7\)
Substitute the values into the formula: \(G=\frac{5\times7+4\times(-2)}{5 + 4}\)

Step3: Calculate the numerator and denominator

First, calculate the numerator: \(5\times7+4\times(-2)=35-8 = 27\)
The denominator \(m + n=5 + 4=9\)

Step4: Find the value of \(G\)

\(G=\frac{27}{9}=3\) (Wait, there was a mistake above. Let's use another way. The length of the segment \(DF\) is \(7-(-2)=9\) units. Since the ratio \(DG:GF = 5:4\), the length of \(DG=\frac{5}{5 + 4}\times9 = 5\) units. Starting from \(D=-2\), moving \(5\) units to the right: \(-2+5 = 3\))

So the location of point \(G\) is \(3\)

Answer:

\(2\)