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 the 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)
Identify the given ratio as m : n = 3 : 5 , so m = 3 and n = 5 .
Determine the coordinates of the endpoints: x 1 = − 14 and x 2 = 2 .
Substitute the values into the formula: ( m + n m ) ( x 2 − x 1 ) + x 1 = ( 3 + 5 3 ) ( 2 − ( − 14 )) + ( − 14 ) .
The correct expression is: ( 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 line segment QS in a 3 : 5 ratio. We need to find the expression that correctly uses the formula ( m + n m ) ( x 2 − x 1 ) + x 1 to find the location of point R .
Identify Variables In the given ratio 3 : 5 , we have m = 3 and n = 5 . The coordinates of the endpoints are x 1 = − 14 (coordinate of Q ) and x 2 = 2 (coordinate of S ).
Substitute Values into Formula Now, we substitute these values into the formula: ( m + n m ) ( x 2 − x 1 ) + x 1 = ( 3 + 5 3 ) ( 2 − ( − 14 )) + ( − 14 ) This simplifies to: ( 8 3 ) ( 2 + 14 ) − 14 = ( 8 3 ) ( 16 ) − 14
Determine Correct Expression The expression that correctly uses the formula is: ( 3 + 5 3 ) ( 2 − ( − 14 )) + ( − 14 ) Comparing this with the given options, we find that it matches the first option.
Final Answer Therefore, 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., commercial and residential) according to a specific ratio, you can use this formula to determine the exact point where the division should occur. For instance, if a street is 1000 meters long and you want to divide it in a 2:3 ratio, this formula helps you find the precise location for the boundary between the two sections, ensuring fair and proportional allocation of space.
The correct expression to find the location of point R is option A: ( 3 + 5 3 ) ( 2 − ( − 14 )) + ( − 14 ) . This uses the provided ratio and endpoints appropriately in the formula.
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