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)
Determine the ratio m : n as 1 : 2 since P is 3 1 of the way from A to B .
Substitute the coordinates of points A ( 1 , 5 ) and B ( − 5 , − 7 ) , and the ratio m : n = 1 : 2 into the formula for the x -coordinate: x = ( 1 + 2 1 ) ( − 5 − 1 ) + 1 .
Substitute the coordinates of points A ( 1 , 5 ) and B ( − 5 , − 7 ) , and the ratio m : n = 1 : 2 into the formula for the y -coordinate: y = ( 1 + 2 1 ) ( − 7 − 5 ) + 5 .
Calculate the x and y coordinates, resulting in the point P ( − 1 , 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 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 m : n is the ratio in which point P divides the line segment A B .
Substitute the values In this case, we have A ( 1 , 5 ) , B ( − 5 , − 7 ) , and the ratio m : n = 1 : 2 . So, x 1 = 1 , y 1 = 5 , x 2 = − 5 , y 2 = − 7 , m = 1 , and n = 2 .
Now, we can substitute these values into the formulas for the x and y coordinates of point P :
x = ( 1 + 2 1 ) ( − 5 − 1 ) + 1 y = ( 1 + 2 1 ) ( − 7 − 5 ) + 5
Calculate x and y coordinates Let's calculate the x -coordinate:
x = ( 3 1 ) ( − 6 ) + 1 x = − 2 + 1 x = − 1
Now, let's calculate the y -coordinate:
y = ( 3 1 ) ( − 12 ) + 5 y = − 4 + 5 y = 1
Final Answer Therefore, the coordinates of point P are ( − 1 , 1 ) .
Examples
In computer graphics, when drawing a line between two points, you might want to find a point that's a certain fraction of the way along that line. This is useful for creating smooth animations or placing objects at specific positions along a path. The formula we used helps calculate the exact coordinates of that intermediate point.
The coordinates of point P, which divides the segment from A to B in the ratio 1:2, are (-1, 1). This calculation is done using the given formulas for the coordinates based on the determined ratio. Hence, point P is located at (-1, 1).
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