Directed line segment from $A$ to $B$ such that $P$ is $\frac{2}{3}$ the length of the line segment from $A$ to $B$?
$x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$
$v=\left(\frac{m}{m+n}\right)\left(v_2-v_1\right)+v_1$
(2,-1)
(4,-3)
(-1,2)
(3,-2)
Answer (2)
Identify the coordinates of points A and B, and the ratio in which point P divides the line segment AB.
Substitute the given values into the formula for the x-coordinate of P and calculate the result: x = 3 10 .
Substitute the given values into the formula for the y-coordinate of P and calculate the result: v = − 3 7 .
State the coordinates of point P: ( 3 10 , − 3 7 ) .
Explanation
Identify given values and formulas. We are given the coordinates of point A as (2, -1) and point B as (4, -3). We are also given the formulas to find the coordinates of point P, which divides the line segment AB in the ratio 2:1. The formulas are:
x = ( m + n m ) ( x 2 − x 1 ) + x 1 v = ( m + n m ) ( v 2 − v 1 ) + v 1
Here, ( x 1 , v 1 ) = ( 2 , − 1 ) , ( x 2 , v 2 ) = ( 4 , − 3 ) , m = 2 , and n = 1 .
Calculate the x-coordinate of P. Now, we substitute the given values into the formula for the x-coordinate:
x = ( 2 + 1 2 ) ( 4 − 2 ) + 2 x = ( 3 2 ) ( 2 ) + 2 x = 3 4 + 2 x = 3 4 + 3 6 x = 3 10
Calculate the y-coordinate of P. Next, we substitute the given values into the formula for the y-coordinate:
v = ( 2 + 1 2 ) ( − 3 − ( − 1 )) + ( − 1 ) v = ( 3 2 ) ( − 3 + 1 ) − 1 v = ( 3 2 ) ( − 2 ) − 1 v = − 3 4 − 1 v = − 3 4 − 3 3 v = − 3 7
State the coordinates of P. Therefore, the coordinates of point P are ( 3 10 , − 3 7 ) .
Examples
In computer graphics, when drawing a line, you might want to find a point that is a certain fraction of the way along that line. This is useful for creating gradients, animations, or other visual effects. The formula we used helps to calculate the exact coordinates of that point.
To find point P that divides the segment from A(2, -1) to B(4, -3) in a 2:1 ratio, we used the formulas for the coordinates of P. The calculations yield the coordinates of point P as ( 3 10 , − 3 7 ) .
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