Question

what is the location of point g, which partitions the directed line segment from f to d into an 8 : 5 ratio? (image of number line with d at -4 and f at 9, and four options: 1, 4, 3, 2)

Explanation:

Step1: Identify coordinates of F and D

From the number line, \( F = 9 \) and \( D=-4 \). The ratio is \( m:n = 8:5 \) (from F to D, so F is the starting point, D is the ending point).

Step2: Use section formula for a line segment

The formula to find a point \( G \) that divides the line segment from \( F(x_1) \) to \( D(x_2) \) in the ratio \( m:n \) is:
\( G = x_1+\frac{m}{m + n}(x_2 - x_1) \)

Substitute \( x_1 = 9 \), \( x_2=-4 \), \( m = 8 \), \( n = 5 \):
First, calculate \( x_2 - x_1=-4 - 9=-13 \)
Then, \( \frac{m}{m + n}=\frac{8}{8 + 5}=\frac{8}{13} \)
Multiply: \( \frac{8}{13}\times(-13)=-8 \)
Add to \( x_1 \): \( 9+(-8)=1 \)

Answer:

1