What are the $x$- and $y$-coordinates of point P on the directed line segment from $A$ to $B$ such that $P$ is $\frac{1}{3}$ the length of the line segment from $A$ to $B$?
$x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$
$y=\left(\frac{m}{m+n}\right)\left(y_2-y_1\right)+y_1$
(1,5)
(0,3)
(-4,-5)
(-5,-7)
Answer (2)
Point P divides the line segment A B in the ratio 1 : 2 .
Apply the section formula for the x-coordinate: x = ( 3 1 ) ( − 5 − 1 ) + 1 = − 1 .
Apply the section formula for the y-coordinate: y = ( 3 1 ) ( − 7 − 5 ) + 5 = 1 .
The coordinates of point P are ( − 1 , 1 ) .
Explanation
Problem Analysis and Setup We are given two points, A ( 1 , 5 ) and B ( − 5 , − 7 ) , and we want to find the coordinates of point P on the directed line segment from A to B such that P is 3 1 the length of the line segment from A to B . This means that the ratio of A P to PB is 1 : 2 . We are also given the formulas for finding the coordinates of P :
x = ( m + n m ) ( x 2 − x 1 ) + x 1 y = ( m + n m ) ( y 2 − y 1 ) + y 1
where ( x 1 , y 1 ) are the coordinates of point A , ( x 2 , y 2 ) are the coordinates of point B , and m : n is the ratio in which P divides the line segment A B . In our case, m = 1 and n = 2 .
Calculate the x-coordinate We have A ( 1 , 5 ) and B ( − 5 , − 7 ) , so x 1 = 1 , y 1 = 5 , x 2 = − 5 , and y 2 = − 7 . Also, m = 1 and n = 2 . Substituting these values into the formula for the x -coordinate of P , we get:
x = ( 1 + 2 1 ) ( − 5 − 1 ) + 1 = ( 3 1 ) ( − 6 ) + 1 = − 2 + 1 = − 1
Calculate the y-coordinate Now, substituting the values into the formula for the y -coordinate of P , we get:
y = ( 1 + 2 1 ) ( − 7 − 5 ) + 5 = ( 3 1 ) ( − 12 ) + 5 = − 4 + 5 = 1
State the coordinates of P Therefore, the coordinates of point P are ( − 1 , 1 ) .
Examples
Imagine you're designing a video game where a character needs to move from point A to point B, but only covers 1/3 of the distance in a single step. Using the section formula, you can easily calculate the exact coordinates where the character will land after that first move. This ensures precise and predictable movements within the game environment.
The coordinates of point P on the directed line segment from A(1, 5) to B(-5, -7) that is 3 1 of the distance from A to B are ( − 1 , 1 ) .
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