Question

select the coordinates of point x on rs that is \\(\frac{1}{6}\\) of the distance from r to s.\
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a) \\((3, -1\frac{1}{3})\\)\
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b) \\((-3, 1\frac{1}{3})\\)

Explanation:

Step1: Recall the section formula

The coordinates of a point \( X(x,y) \) that divides the line segment joining \( R(x_1,y_1) \) and \( S(x_2,y_2) \) in the ratio \( m:n \) (from \( R \) to \( S \)) are given by \( x=\frac{mx_2 + nx_1}{m + n} \) and \( y=\frac{my_2 + ny_1}{m + n} \). Here, \( m = 1 \), \( n=5 \) (since the point is \( \frac{1}{6} \) of the distance from \( R \) to \( S \), so the ratio \( m:n=1:5 \)), \( R(-4,2) \) so \( x_1=-4,y_1 = 2 \), and \( S(2,-2) \) so \( x_2 = 2,y_2=-2 \).

Step2: Calculate the x - coordinate of X

Substitute the values into the formula for \( x \):
\( x=\frac{1\times2+5\times(-4)}{1 + 5}=\frac{2-20}{6}=\frac{-18}{6}=-3 \)

Step3: Calculate the y - coordinate of X

Substitute the values into the formula for \( y \):
\( y=\frac{1\times(-2)+5\times2}{1 + 5}=\frac{-2 + 10}{6}=\frac{8}{6}=\frac{4}{3}=1\frac{1}{3} \)

Answer:

B) \((-3,1\frac{1}{3})\)