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 $B$?
$x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$
$y=\left(\frac{m}{m+n}\right)\left(y_2-y_1\right)+y_1$
(2,-1)
(4,-3)
(-1,2)
(3,-2)
Answer (2)
Determine the ratio m : n as 2 : 1 based on the given fraction 3 2 of the line segment.
Calculate the x -coordinate of point P using the formula x = ( m + n m ) ( x 2 − x 1 ) + x 1 , which gives x = 3 8 .
Calculate the y -coordinate of point P using the formula y = ( m + n m ) ( y 2 − y 1 ) + y 1 , which gives y = − 3 5 .
State the coordinates of point P as ( 3 8 , − 3 5 ) .
Explanation
Problem Analysis and Setup We are given two points, A ( 2 , − 1 ) and B ( 3 , − 2 ) , and 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 . This means that the ratio A P : PB = 2 : 1 , so m = 2 and n = 1 . 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
Calculate the x-coordinate First, let's find the x -coordinate of point P . We have m = 2 , n = 1 , x 1 = 2 , and x 2 = 3 . Plugging these values into the formula, we get:
x = ( 2 + 1 2 ) ( 3 − 2 ) + 2 = ( 3 2 ) ( 1 ) + 2 = 3 2 + 2 = 3 2 + 3 6 = 3 8
Calculate the y-coordinate Now, let's find the y -coordinate of point P . We have m = 2 , n = 1 , y 1 = − 1 , and y 2 = − 2 . Plugging these values into the formula, we get:
y = ( 2 + 1 2 ) ( − 2 − ( − 1 )) + ( − 1 ) = ( 3 2 ) ( − 2 + 1 ) − 1 = ( 3 2 ) ( − 1 ) − 1 = − 3 2 − 1 = − 3 2 − 3 3 = − 3 5
State the coordinates of P Therefore, the coordinates of point P are ( 3 8 , − 3 5 ) .
Final Answer The x -coordinate of point P is 3 8 and the y -coordinate of point P is − 3 5 .
Examples
In computer graphics, when drawing a line between two points, you might want to find a point that is a certain fraction of the way along that line. This is useful for creating smooth animations or for placing objects at specific locations along a path. For example, if you have two points defining a line segment and you want to place an object two-thirds of the way along that line, you can use the formula we just used to find the exact coordinates of that point.
The coordinates of point P, which is 3 2 of the way from point A(2, -1) to point B(3, -2), are oxed{\left(\frac{8}{3}, -\frac{5}{3}\right)} .
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