Line segment $P R$ is a directed line segment beginning at $P(-10,7)$ and ending at $R(8,-5)$. Find point Q on the line segment PR that partitions it into the segments PQ and QR in the ratio 4:5.
A. $\left(-2, \frac{5}{3}\right)$
B. $\left(-2,-\frac{5}{3}\right)$
C. $\left(-\frac{9}{2}, 3\right)$
D. $\left(0,-\frac{1}{3}\right)$
Answer (1)
Use the section formula to find the coordinates of point Q .
Substitute the given values P ( − 10 , 7 ) , R ( 8 , − 5 ) , m = 4 , and n = 5 into the section formula.
Calculate the x-coordinate of Q : x = 4 + 5 4 ( 8 ) + 5 ( − 10 ) = − 2 .
Calculate the y-coordinate of Q : y = 4 + 5 4 ( − 5 ) + 5 ( 7 ) = 3 5 .
The coordinates of point Q are ( − 2 , 3 5 ) .
Explanation
Problem Analysis We are given a directed line segment PR with P ( − 10 , 7 ) and R ( 8 , − 5 ) . We need to find a point Q on this line segment such that the ratio PQ : QR = 4 : 5 . This is a section formula problem.
Section Formula The section formula states that if a point Q ( x , y ) divides the line segment joining P ( x 1 , y 1 ) and R ( x 2 , y 2 ) in the ratio m : n , then the coordinates of Q are given by:
x = m + n m x 2 + n x 1 y = m + n m y 2 + n y 1
In our case, P ( − 10 , 7 ) , R ( 8 , − 5 ) , m = 4 , and n = 5 .
Calculating Coordinates Now, we substitute the given values into the section formula to find the coordinates of point Q .
For the x-coordinate: x = 4 + 5 4 ( 8 ) + 5 ( − 10 ) = 9 32 − 50 = 9 − 18 = − 2
For the y-coordinate: y = 4 + 5 4 ( − 5 ) + 5 ( 7 ) = 9 − 20 + 35 = 9 15 = 3 5
Therefore, the coordinates of point Q are ( − 2 , 3 5 ) .
Final Answer The coordinates of point Q are ( − 2 , 3 5 ) . Comparing this with the given options, we find that option A matches our calculated coordinates.
Examples
Imagine you're planning a road trip and need to meet a friend along the way. You want to choose a meeting point that's a specific fraction of the total distance. Using the section formula, you can calculate the exact coordinates (or location) of that meeting point, ensuring a fair division of the journey for both of you. For instance, if you want to meet 4/9 of the way, you can use the formula to find the precise location on the map.
