Line segment $PQ$ is a directed line segment beginning at $P(6, -5)$ and ending at $Q(-2, 4)$. Find point $R$ on the line segment $PQ$ that partitions it into the segments $PR$ and $RQ$ in the ratio $3:2$.
A. $\left(\frac{6}{5}, \frac{2}{5}\right)$
B. $\left(\frac{14}{5},-\frac{7}{5}\right)$
C. $\left(-\frac{6}{6}, \frac{2}{5}\right)$
D. $\left(\frac{14}{6}, \frac{7}{5}\right)$
Answer (2)
Use the section formula to find the coordinates of point R that divides the line segment PQ in the ratio 3 : 2 .
Substitute the coordinates of points P and Q and the ratio m : n into the section formula.
Calculate the x and y coordinates of point R .
The coordinates of point R are ( 5 6 , 5 2 ) .
Explanation
Problem Analysis We are given a directed line segment PQ with P ( 6 , − 5 ) and Q ( − 2 , 4 ) . We need to find the point R on the line segment PQ that divides it in the ratio 3 : 2 . This means that the ratio of the length of PR to the length of RQ is 3 : 2 , or RQ PR = 2 3 .
Applying the Section Formula To find the coordinates of point R , we can use the section formula. The section formula states that if a point R ( x , y ) divides the line segment joining P ( x 1 , y 1 ) and Q ( x 2 , y 2 ) in the ratio m : n , then the coordinates of R are given by:
x = m + n m x 2 + n x 1 y = m + n m y 2 + n y 1
In this case, we have P ( 6 , − 5 ) , Q ( − 2 , 4 ) , m = 3 , and n = 2 .
Calculating the Coordinates of Point R Now, we substitute the given values into the section formula to find the coordinates of point R :
x = 3 + 2 3 ( − 2 ) + 2 ( 6 ) = 5 − 6 + 12 = 5 6 = 1.2 y = 3 + 2 3 ( 4 ) + 2 ( − 5 ) = 5 12 − 10 = 5 2 = 0.4
So, the coordinates of point R are ( 5 6 , 5 2 ) .
Final Answer Therefore, the point R that divides the line segment PQ in the ratio 3 : 2 is ( 5 6 , 5 2 ) .
Examples
In architecture, when designing a staircase, you might need to divide a certain length into specific ratios to place the steps correctly. This problem demonstrates how to find a point that divides a line segment in a given ratio, which can be applied to ensure each step is evenly spaced and the staircase is aesthetically pleasing and structurally sound. Similarly, in urban planning, this concept can be used to allocate resources or design infrastructure along a road or path, ensuring fair distribution and optimal placement.
Point R divides the line segment PQ in the ratio of 3:2, which can be calculated using the section formula. The coordinates of point R are ( 5 6 , 5 2 ) . Thus, the correct answer is option A.
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