Question

point r (\boxed{ }, \boxed{ }) is $\frac{1}{4}$ the distance from s to t. write your answer in decimal form.

s (-6,6) \bullet

t (6,-2) \bullet

Explanation:

Step1: Recall the section formula

To find a point \( R(x,y) \) that divides the line segment joining \( S(x_1,y_1) \) and \( T(x_2,y_2) \) in the ratio \( m:n \), the coordinates of \( R \) are given by:
\[
x=\frac{mx_2 + nx_1}{m + n}, \quad y=\frac{my_2 + ny_1}{m + n}
\]
Here, \( R \) is \( \frac{1}{4} \) the distance from \( S \) to \( T \), so the ratio \( m:n = 1:3 \) (since \( R \) divides \( ST \) such that \( SR:RT = 1:3 \)). \( S(-6,6) \) so \( x_1=-6,y_1 = 6 \), and \( T(6,-2) \) so \( x_2 = 6,y_2=-2 \), \( m = 1 \), \( n = 3 \).

Step2: Calculate the x - coordinate of R

Substitute the values into the formula for \( x \):
\[
x=\frac{1\times6+3\times(-6)}{1 + 3}=\frac{6-18}{4}=\frac{-12}{4}=-3.0
\]

Step3: Calculate the y - coordinate of R

Substitute the values into the formula for \( y \):
\[
y=\frac{1\times(-2)+3\times6}{1 + 3}=\frac{-2 + 18}{4}=\frac{16}{4}=4.0
\]

Answer:

\( R(-3.0, 4.0) \)