Question

practice with directed line segments.
image: number line with d at -4, e at 5
what is the location of point f, which partitions the directed line segment from d to e into a 5:6 ratio?
options: $\frac{-1}{11}$, $\frac{1}{11}$, $\frac{2}{15}$, $\frac{15}{2}$

Explanation:

Step1: Identify coordinates of D and E

From the number line, \( D = -4 \) and \( E = 5 \). The ratio is \( m:n = 5:6 \).

Step2: Apply section formula

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

Substitute \( x_1=-4 \), \( x_2 = 5 \), \( m = 5 \), \( n = 6 \):

\( F=\frac{5\times5+6\times(-4)}{5 + 6} \)

Step3: Calculate numerator and denominator

Numerator: \( 25-24 = 1 \)

Denominator: \( 11 \)

So \( F=\frac{1}{11} \)

Answer:

\(\frac{1}{11}\) (corresponding to the option \(\boldsymbol{\frac{1}{11}}\))