Question

chris is traveling from the cleaners to the pizza parlor. the cleaners is located at (-6, 4) and the pizza parlor is located at (-8, 8) on a coordinate grid map. chris plans to stop for a break when the ratio of the distance he has traveled to the distance he has left to travel is 4:6. where should he take a break? write your answer as a decimal. chris should take a break at (□, □).

Explanation:

Step1: Understand the section formula

To find a point that divides a line segment in the ratio \( m:n \), we use the section formula. If a point \( (x,y) \) divides the line segment joining \( (x_1,y_1) \) and \( (x_2,y_2) \) in the ratio \( m:n \), then:
\[
x=\frac{mx_2 + nx_1}{m + n}, \quad y=\frac{my_2 + ny_1}{m + n}
\]
Here, the cleaners is \( (x_1,y_1)=(-6,4) \), the pizza parlor is \( (x_2,y_2)=(-8,8) \), and the ratio \( m:n = 4:6 \) (distance traveled : distance left, so the point divides the segment from cleaners to pizza parlor in the ratio \( 4:6 \)).

Step2: Calculate the x - coordinate

First, substitute \( m = 4 \), \( n = 6 \), \( x_1=-6 \), \( x_2=-8 \) into the formula for \( x \):
\[
x=\frac{4\times(-8)+6\times(-6)}{4 + 6}=\frac{-32-36}{10}=\frac{-68}{10}=-6.8
\]

Step3: Calculate the y - coordinate

Substitute \( m = 4 \), \( n = 6 \), \( y_1 = 4 \), \( y_2 = 8 \) into the formula for \( y \):
\[
y=\frac{4\times8+6\times4}{4 + 6}=\frac{32 + 24}{10}=\frac{56}{10}=5.6
\]

Answer:

Chris should take a break at \((-6.8, 5.6)\)