On a number line, the directed line segment from [tex]$Q$[/tex] to [tex]$S$[/tex] has endpoints [tex]$Q$[/tex] at -8 and [tex]$S$[/tex] at 12. Point [tex]$R$[/tex] partitions the directed line segment from [tex]$Q$[/tex] to [tex]$S$[/tex] in a 4:1 ratio. Which expression correctly uses the formula [tex]$\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$[/tex] to find the location of point R?
A. [tex]$\left(\frac{1}{1+4}\right)(12-(-8))+(-8)$[/tex]
B. [tex]$\left(\frac{4}{4+1}\right)(12-(-8))+(-8)$[/tex]
C. [tex]$\left(\frac{4}{4+1}\right)(-8-12)+12$[/tex]
D. [tex]$\left(\frac{4}{1+4}\right)(-8-12)+12$[/tex]
Answer (2)
Identify the coordinates of the endpoints of the directed line segment: x 1 = − 8 and x 2 = 12 .
Determine the ratio m : n as 4 : 1 , so m = 4 and n = 1 .
Substitute the values into the partitioning formula: ( m + n m ) ( x 2 − x 1 ) + x 1 = ( 4 + 1 4 ) ( 12 − ( − 8 ) ) + ( − 8 ) .
The correct expression is: ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 ) .
Explanation
Problem Analysis Let's analyze the problem. We are given a directed line segment from point Q to point S on a number line. The coordinate of point Q is -8, and the coordinate of point S is 12. Point R partitions the segment QS in a 4:1 ratio. We are given the formula to find the location of point R: ( m + n m ) ( x 2 − x 1 ) + x 1 . Our objective is to determine the correct expression using the given formula to find the location of point R.
Identify x1 and x2 First, we need to identify x 1 and x 2 . Since the segment is directed from Q to S, x 1 corresponds to the coordinate of Q, and x 2 corresponds to the coordinate of S. Therefore, x 1 = − 8 and x 2 = 12 .
Identify m and n Next, we need to identify m and n . The ratio is given as 4:1, so m = 4 and n = 1 .
Substitute values into the formula Now, we substitute the values of m , n , x 1 , and x 2 into the formula ( m + n m ) ( x 2 − x 1 ) + x 1 . This gives us: ( 4 + 1 4 ) ( 12 − ( − 8 ) ) + ( − 8 )
Identify the correct expression Finally, we compare the derived expression with the given options to identify the correct one. The correct expression is: ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 )
Examples
In city planning, determining the location of a new facility (like a park or a school) along a road segment to serve two neighborhoods in a specific ratio is a practical application of partitioning a line segment. The formula helps ensure equitable access based on the population ratio of the neighborhoods.
The expression that finds the location of point R, which partitions the segment from Q to S in a 4:1 ratio, is ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 ) . Thus, option B is the correct answer. It applies the partitioning formula correctly using the coordinates of Q and S.
;
