On a number line, the directed line segment from Q to S has endpoints Q at -14 and S at 2. Point R 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 [tex]$R$[/tex]?
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 coordinates of the endpoints Q and S as x 1 = − 14 and x 2 = 2 , respectively.
Determine the ratio m : n = 3 : 5 , so m = 3 and n = 5 .
Substitute the values into the formula ( m + n m ) ( x 2 − x 1 ) + x 1 to get ( 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 directed line segment from Q to S in a 3:5 ratio. We are asked 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 and substitute into the formula. Here, x 1 is the coordinate of point Q, so x 1 = − 14 . And x 2 is the coordinate of point S, so x 2 = 2 . The ratio is m : n = 3 : 5 , so m = 3 and n = 5 . Substituting these values into the formula ( m + n m ) ( x 2 − x 1 ) + x 1 , we get the expression ( 3 + 5 3 ) ( 2 − ( − 14 ) ) + ( − 14 ) .
Simplify the expression. The expression is ( 3 + 5 3 ) ( 2 − ( − 14 ) ) + ( − 14 ) . Simplifying this expression, we get ( 8 3 ) ( 2 + 14 ) − 14 = ( 8 3 ) ( 16 ) − 14 = 3 ( 2 ) − 14 = 6 − 14 = − 8 .
Compare with the given options. Comparing the derived expression ( 3 + 5 3 ) ( 2 − ( − 14 ) ) + ( − 14 ) with the given options, we see 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., residential and commercial) based on a specific ratio, you can use this formula to find 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, you can use this formula to find the point that separates the two sections. This ensures fair and proportional allocation of resources or space.
The correct expression to find the location of point R using the given formula is ( 3 + 5 3 ) ( 2 − ( − 14 )) + ( − 14 ) . This corresponds to option A. By evaluating the endpoints Q and S, we accurately apply the given ratio to compute the position of R.
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