On a number line, the directed line segment from [tex]$Q$[/tex] to [tex]$S$[/tex] has endpoints [tex]$Q$[/tex] at -14 and [tex]$S$[/tex] at 2. Point [tex]$R$[/tex] partitions directed line segment from [tex]$Q$[/tex] to [tex]$S$[/tex] in a 3:5 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]$\left(\frac{3}{3+5}\right)(2-(-14))+(-14)$[/tex]
B. [tex]$\left(\frac{3}{3+5}\right)(-14-2)+2$[/tex]
C. [tex]$\left(\frac{3}{3+5}\right)(2-14)+14$[/tex]
D. [tex]$\left(\frac{3}{3+5}\right)(-14-2)-2$[/tex]
Answer (2)
The problem involves finding the correct expression to determine the location of a point R that partitions a directed line segment QS in a given ratio. The formula ( m + n m ) ( x 2 − x 1 ) + x 1 is used, where x 1 and x 2 are the coordinates of points Q and S respectively, and m : n is the given ratio. By substituting the given values x 1 = − 14 , x 2 = 2 , m = 3 , and n = 5 into the formula, we find the correct expression. ( 3 + 5 3 ) ( 2 − ( − 14 )) + ( − 14 ) .
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 -14, and the coordinate of point S is 2. Point R partitions the segment QS in a 3 : 5 ratio. We need 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 represents the coordinate of the starting point Q , and x 2 represents the coordinate of the ending point S . The ratio is given as m : n , where m = 3 and n = 5 . Therefore, x 1 = − 14 and x 2 = 2 .
Substitute Values Substitute the values into the formula: ( m + n m ) ( x 2 − x 1 ) + x 1 = ( 3 + 5 3 ) ( 2 − ( − 14 )) + ( − 14 ) .
Simplify Simplify the expression: ( 3 + 5 3 ) ( 2 − ( − 14 )) + ( − 14 ) = ( 8 3 ) ( 2 + 14 ) − 14 = ( 8 3 ) ( 16 ) − 14 = 3 × 2 − 14 = 6 − 14 = − 8 .
Final Answer The correct expression is ( 3 + 5 3 ) ( 2 − ( − 14 )) + ( − 14 ) .
Examples
In city planning, if you need to divide a street into sections for different purposes (e.g., residential and commercial) based on a specific ratio, you can use this formula to determine the exact location where one section ends and another begins. For example, if a street is 1000 meters long and you want to divide it in a 2:3 ratio, you can use this formula to find the point that separates the two sections.
The correct expression to find point R that partitions the segment from Q to S in a 3:5 ratio is option A: ( 3 + 5 3 ) ( 2 − ( − 14 )) + ( − 14 ) .
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