What are the $x$- and $y$-coordinates of point P on the directed line segment from $A$ to $B$ such that $P$ is $\frac{2}{3}$ the length of the line segment from $A$ to $B$?
[tex]x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1[/tex]
[tex]v=\left(\frac{m}{m+n}\right)\left(v_2-v_1\right)+v_1[/tex]
$A(2,-1)$
$B(4,-3)$
A. $(-1,2)$
B. $(3,-2)$
Answer (2)
Identify the coordinates of points A and B, and the ratio m:n representing the fraction of the distance from A to B.
Substitute the values into the formula for the x-coordinate of point P: x = ( m + n m ) ( x 2 − x 1 ) + x 1 .
Substitute the values into the formula for the y-coordinate of point P: y = ( m + n m ) ( y 2 − y 1 ) + y 1 .
Calculate the coordinates of point P: ( 3 10 , − 3 7 ) .
Explanation
Problem Analysis and Setup We are given two points, A ( 2 , − 1 ) and B ( 4 , − 3 ) . We want to find the coordinates of point P on the directed line segment from A to B such that P is 3 2 the length of the line segment from A to B . We are also given the formulas for the x and y coordinates of point P :
x = ( m + n m ) ( x 2 − x 1 ) + x 1 y = ( m + n m ) ( y 2 − y 1 ) + y 1
Here, ( x 1 , y 1 ) = ( 2 , − 1 ) and ( x 2 , y 2 ) = ( 4 , − 3 ) . Since P is 3 2 the length of the line segment from A to B , the ratio m : n is 2 : 1 , so m = 2 and n = 1 .
Calculate x-coordinate Now, we substitute the given values into the formula for the x -coordinate:
x = ( 2 + 1 2 ) ( 4 − 2 ) + 2 x = ( 3 2 ) ( 2 ) + 2 x = 3 4 + 2 x = 3 4 + 3 6 x = 3 10
Calculate y-coordinate Next, we substitute the given values into the formula for the y -coordinate:
y = ( 2 + 1 2 ) ( − 3 − ( − 1 )) + ( − 1 ) y = ( 3 2 ) ( − 3 + 1 ) − 1 y = ( 3 2 ) ( − 2 ) − 1 y = − 3 4 − 1 y = − 3 4 − 3 3 y = − 3 7
Final Answer Therefore, the coordinates of point P are ( 3 10 , − 3 7 ) .
Examples
In computer graphics, when drawing a line from one point to another, you might want to place an object at a specific fraction along that line. This problem demonstrates how to calculate the exact coordinates of that object's position. For example, if you're creating a game and want an enemy to appear two-thirds of the way along a path between two towers, you'd use this formula to find the enemy's coordinates.
The coordinates of point P on the directed line segment from A(2,-1) to B(4,-3), located at 3 2 of the way from A to B, are ( 3 10 , − 3 7 ) .
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