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 $B$?
$x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$
$y=\left(\frac{m}{m+n}\right)\left(y_2-y_1\right)+y_1$
$A(1,5)$
$B(-5,-7)$
Answer (1)
The problem asks to find the coordinates of a point P that is 1/3 of the way from point A to point B.
The ratio m:n is determined to be 1:2, representing the fraction of the distance from A to B.
The formulas for the x and y coordinates are used to calculate the coordinates of point P.
The coordinates of point P are found to be ( − 1 , 1 ) .
Explanation
Problem Analysis and Given Information We are given two points, A ( 1 , 5 ) and B ( − 5 , − 7 ) , and we want to find the coordinates of point P that lies 3 1 of the way from A to B . We are also given the formulas to calculate 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
where ( x 1 , y 1 ) are the coordinates of point A , ( x 2 , y 2 ) are the coordinates of point B , and the ratio m : n represents the fraction of the distance from A to B at which point P is located.
Determine the Ratio and Substitute Values Since point P is 3 1 of the way from A to B , the ratio m : n is 1 : 2 . This means that m = 1 and n = 2 .
Now, we can substitute the given values into the formulas for the x and y coordinates:
x = ( 1 + 2 1 ) ( − 5 − 1 ) + 1 y = ( 1 + 2 1 ) ( − 7 − 5 ) + 5
Calculate x and y Coordinates Now, we can calculate the x -coordinate:
x = ( 3 1 ) ( − 6 ) + 1 x = − 2 + 1 x = − 1
And the y -coordinate:
y = ( 3 1 ) ( − 12 ) + 5 y = − 4 + 5 y = 1
State the Coordinates of Point P Therefore, the coordinates of point P are ( − 1 , 1 ) .
Examples
In computer graphics, when drawing a line from one point to another, you might want to find a point that is a certain fraction of the way along that line. This is useful for creating animations or drawing dashed lines. For example, if you have a line from point A to point B, and you want to draw a dash that is 1/3 of the way along the line, you would use the formula we just used to find the coordinates of that point.
