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 Q and S as x 1 = − 8 and x 2 = 12 , respectively.
Determine the ratio m : n as 4 : 1 , so m = 4 and n = 1 .
Substitute the values into the formula ( m + n m ) ( x 2 − x 1 ) + x 1 to get ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 ) .
The correct expression is ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 ) .
Explanation
Problem Analysis 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 asked to find the correct expression using the formula ( m + n m ) ( x 2 − x 1 ) + x 1 to find the location of point R .
Identify Variables In the given formula, x 1 and x 2 are the coordinates of the endpoints of the directed line segment, and m and n define the ratio in which point R partitions the segment. Since the segment is directed from Q to S , we have x 1 = − 8 (coordinate of Q ) and x 2 = 12 (coordinate of S ). The ratio is 4 : 1 , so m = 4 and n = 1 .
Substitute Values into 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 = ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 ) This simplifies to: ( 5 4 ) ( 12 + 8 ) − 8 = ( 5 4 ) ( 20 ) − 8 So the correct expression is ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 ) .
Identify Correct Expression Comparing the expression we derived with the given options, we find that the correct expression is ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 ) .
Final Answer The correct expression to find the location of point R is ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 ) .
Examples
In city planning, determining the location of a new facility (like a bus stop or a park) often involves dividing the distance between two key points in a specific ratio to serve different neighborhoods effectively. For instance, if we want to place a park at a location that's 4/5 of the distance from a residential area to a commercial hub, we use the formula for partitioning a line segment to find the exact spot. This ensures the park is conveniently located for both residents and workers, enhancing community accessibility and promoting balanced urban development.
To find the location of point R that partitions the segment from Q (-8) to S (12) in a 4:1 ratio, we use the formula which leads us to the expression: ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 ) . The correct multiple-choice option is B.
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