Question

given df with d(-1, 11) and f(-9, -5), if e partitions df such that the ratio of de to df is 5:8, find the coordinates of e.

Explanation:

Step1: Recall the section formula

The section formula for a point \( E(x,y) \) that divides the line segment joining \( D(x_1,y_1) \) and \( F(x_2,y_2) \) in the ratio \( m:n \) is given by:
\[
x=\frac{mx_2 + nx_1}{m + n}, \quad y=\frac{my_2 + ny_1}{m + n}
\]
Here, \( D(-1,11) \), \( F(-9,-5) \), and the ratio \( DE:DF = 5:8 \), so \( DE:EF=5:(8 - 5)=5:3 \), thus \( m = 5 \), \( n = 3 \), \( x_1=-1 \), \( y_1 = 11 \), \( x_2=-9 \), \( y_2=-5 \).

Step2: Calculate the x - coordinate of E

Substitute the values into the formula for \( x \):
\[
x=\frac{5\times(-9)+3\times(-1)}{5 + 3}=\frac{-45-3}{8}=\frac{-48}{8}=-6
\]

Step3: Calculate the y - coordinate of E

Substitute the values into the formula for \( y \):
\[
y=\frac{5\times(-5)+3\times11}{5 + 3}=\frac{-25 + 33}{8}=\frac{8}{8}=1
\]

Answer:

The coordinates of \( E \) are \( (-6,1) \)