Question

question 18: if m(2, -7) and z(6, 3), calculate the coordinates of point f which is ¼ of the distance from m to z on \\(\overline{mz}\\). show your algebraic thinking to earn full credit.

Explanation:

Step1: Recall the section formula

The section formula for a point \( F(x,y) \) that divides the line segment joining \( M(x_1,y_1) \) and \( Z(x_2,y_2) \) 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, \( F \) is \( \frac{1}{4} \) of the distance from \( M \) to \( Z \), so the ratio \( m:n = 1:3 \) (since \( F \) divides \( MZ \) such that \( MF: FZ=1:3 \)).
Given \( M(2,-7) \) so \( x_1 = 2,y_1=-7 \) and \( Z(6,3) \) so \( x_2 = 6,y_2 = 3 \), \( m = 1 \), \( n=3 \).

Step2: Calculate the x - coordinate of F

Substitute the values into the formula for \( x \):
\[
x=\frac{1\times6+3\times2}{1 + 3}=\frac{6 + 6}{4}=\frac{12}{4}=3
\]

Step3: Calculate the y - coordinate of F

Substitute the values into the formula for \( y \):
\[
y=\frac{1\times3+3\times(-7)}{1 + 3}=\frac{3-21}{4}=\frac{-18}{4}=-\frac{9}{2}=-4.5
\]

Answer:

The coordinates of point \( F \) are \( (3,-\frac{9}{2}) \) or \( (3, - 4.5) \)