What are the $x$- and $y$-coordinates of point P on the directed line segment from $K$ to $J$ such that $P$ is $\frac{3}{5}$ the length of the line segment from K to J?
$x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$
$v=\left(\frac{m}{m+n}\right)\left(v_2-v_1\right)+v_1$
A. $(40,96)$
B. $(85,105)$
C. $(80,104)$
D. $(96,72)$
Answer (2)
Identified the coordinates of points K and J and the fraction of the distance to point P.
Applied the section formula to calculate the x-coordinate of point P: x = ( 5 3 ) ( 85 − 40 ) + 40 = 67 .
Applied the section formula to calculate the y-coordinate of point P: y = ( 5 3 ) ( 105 − 96 ) + 96 = 101.4 .
Determined the coordinates of point P: ( 67 , 101.4 ) .
Explanation
Problem Analysis and Setup We are given two points, K(40, 96) and J(85, 105), and we want to find the coordinates of point P on the directed line segment from K to J such that P is 5 3 the length of the line segment from K to J. This means that the ratio of KP to KJ is 3:5. 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 K, (x_2, y_2) are the coordinates of point J, and m + n m is the fraction of the distance from K to J. In this case, m + n m = 5 3 .
Substitute the values We are given:
x 1 = 40 y 1 = 96 x 2 = 85 y 2 = 105 m + n m = 5 3
Now, we can substitute these values into the formulas for the x- and y-coordinates of point P.
Calculate x and y coordinates Calculate the x-coordinate of point P:
x = ( 5 3 ) ( 85 − 40 ) + 40 x = ( 5 3 ) ( 45 ) + 40 x = ( 0.6 ) ( 45 ) + 40 x = 27 + 40 x = 67
Calculate the y-coordinate of point P:
y = ( 5 3 ) ( 105 − 96 ) + 96 y = ( 5 3 ) ( 9 ) + 96 y = ( 0.6 ) ( 9 ) + 96 y = 5.4 + 96 y = 101.4
Final Answer Therefore, the coordinates of point P are (67, 101.4).
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 placing objects or creating animations that move along a defined path. For example, if you have a starting point K and an ending point J, and you want to place an object at a point P that is 3/5 of the way from K to J, you would use the section formula to calculate the coordinates of point P.
The coordinates of point P, which is 5 3 of the way from point K to point J, are calculated to be (67, 101.4). However, this does not match any of the provided options. Thus, the correct answer according to calculations does not appear to be listed in the options A through D.
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