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 (1)
Point P divides the directed line segment from A ( 0 , 3 ) to B ( − 5 , − 7 ) in the ratio 1 : 2 .
Apply the section formula: x = ( m + n m ) ( x 2 − x 1 ) + x 1 and y = ( m + n m ) ( y 2 − y 1 ) + y 1 .
Substitute the coordinates of A and B and the ratio m : n = 1 : 2 into the formula.
Calculate the coordinates of point P : ( − 3 5 , − 3 1 ) .
Explanation
Problem Analysis We are given two points, A = ( 0 , 3 ) and B = ( − 5 , − 7 ) . 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 P divides the segment A B in the ratio 1 : 2 . We can use the section formula to find the coordinates of P .
Section Formula The section formula is given by: 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 ) and ( x 2 , y 2 ) are the coordinates of points A and B respectively, and m : n is the ratio in which P divides the segment A B . In our case, ( x 1 , y 1 ) = ( 0 , 3 ) , ( x 2 , y 2 ) = ( − 5 , − 7 ) , and m : n = 1 : 2 .
Calculations Substituting the given values into the section formula, we get: x = ( 1 + 2 1 ) ( − 5 − 0 ) + 0 = ( 3 1 ) ( − 5 ) = − 3 5 y = ( 1 + 2 1 ) ( − 7 − 3 ) + 3 = ( 3 1 ) ( − 10 ) + 3 = − 3 10 + 3 = − 3 10 + 3 9 = − 3 1 So the coordinates of point P are ( − 3 5 , − 3 1 ) .
Final Answer Therefore, the x -coordinate of point P is − 3 5 and the y -coordinate of point P is − 3 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 calculate the exact coordinates where the character will land after that step. This ensures precise and predictable movement within the game environment.
