On a number line, the directed line segment from $Q$ to $S$ has endpoints $Q$ at -8 and $S$ at 12. Point $R$ partitions the directed line segment from $Q$ to $S$ in a 4:1 ratio. Which expression correctly uses the formula $\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$ to find the location of point $R$?
A. $\left(\frac{1}{1+4}\right)(12-(-8))+(-8)$
B. $\left(\frac{4}{4+1}\right)(12-(-8))+(-8)$
C. $\left(\frac{4}{4+1}\right)(-8-12)+12$
D. $\left(\frac{1}{1+4}\right)(-8-(12)+12$
Answer (1)
Identified the coordinates of points Q and S as x 1 = − 8 and x 2 = 12 , respectively.
Recognized the partitioning ratio as m = 4 and n = 1 .
Substituted these values into the section formula: ( m + n m ) ( x 2 − x 1 ) + x 1 .
Determined the correct expression: ( 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 ratio of 4:1. 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 the values First, we need to identify the values of x 1 , x 2 , m , and n from the given information.
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 .
The ratio in which point R partitions the segment is given as 4:1. Thus, m = 4 and n = 1 .
Substitute the 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 = ( 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 the correct expression Comparing the derived expression with the given options, we find that the correct expression is: ( 4 + 1 4 ) ( 12 − ( − 8 )) + ( − 8 )
Therefore, the location of point R is given by this expression.
Examples
In city planning, determining the location of a bus stop along a street segment can be done using ratios. If you want to place a bus stop at a point that divides the street segment in a specific ratio to optimize accessibility for residents, you can use the section formula. For instance, if a street segment is 1000 meters long and you want to place the bus stop 3 2 of the way from one end, you would use a similar formula to find the exact location.
