Question

lm has endpoints l(1, -2) and m(4, -8), and point n on lm partitions the segment such that the ratio of ln to nm is 2:1. what are the coordinates of point n?
a) (-1, -3)
b) (2, -4)
c) (2.5, -5)
d) (3, -6)

Explanation:

Step1: Recall the section formula

The section formula for a point \( N(x,y) \) that divides the line segment joining \( L(x_1,y_1) \) and \( M(x_2,y_2) \) 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, \( L(1,-2) \), \( M(4,-8) \), and the ratio \( LN:NM = 2:1 \), so \( m = 2 \), \( n = 1 \), \( x_1=1 \), \( y_1=-2 \), \( x_2 = 4 \), \( y_2=-8 \).

Step2: Calculate the x - coordinate of N

Substitute the values into the formula for \( x \):

$$ x=\frac{2\times4+1\times1}{2 + 1}=\frac{8 + 1}{3}=\frac{9}{3}=3 $$

Step3: Calculate the y - coordinate of N

Substitute the values into the formula for \( y \):

$$ y=\frac{2\times(-8)+1\times(-2)}{2 + 1}=\frac{-16-2}{3}=\frac{-18}{3}=-6 $$

Answer:

D) \((3,-6)\)