Question

the directed line segment from l to n has endpoints l(-6, 2) and n(5, -3). what are the x- and y-coordinates of point m, which partitions the directed line segment into the ratio 2:5? x = y = -20/7 -4/7 4/7 20/7

Explanation:

Step1: Recall the section formula

The section formula for a point \( M(x, y) \) that divides the line segment joining \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \) is \( x=\frac{mx_2 + nx_1}{m + n} \), \( y=\frac{my_2 + ny_1}{m + n} \). Here, \( (x_1, y_1)=(-6, 2) \), \( (x_2, y_2)=(5, -3) \), \( m = 2 \), \( n = 5 \).

Step2: Calculate the x - coordinate

Substitute the values into the x - coordinate formula:
\( x=\frac{2\times5+5\times(-6)}{2 + 5}=\frac{10-30}{7}=\frac{-20}{7} \)

Step3: Calculate the y - coordinate

Substitute the values into the y - coordinate formula:
\( y=\frac{2\times(-3)+5\times2}{2 + 5}=\frac{-6 + 10}{7}=\frac{4}{7} \)

Answer:

\( x = -\frac{20}{7} \)
\( y=\frac{4}{7} \)