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 ?
$\begin{array}{l}
x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1 \\
y=\left(\frac{m}{m+n}\right)\left(y_2-y_1\right)+y_1
\end{array}$
A. (40,96)
B. (85,105)
C. (80,104)
D. (96,72)
Answer (2)
Calculate the x-coordinate of point P using the section formula: x = ( 5 3 ) ( 80 − 40 ) + 40 = 64 .
Calculate the y-coordinate of point P using the section formula: y = ( 5 3 ) ( 104 − 96 ) + 96 = 100.8 .
Determine the coordinates of point P as ( 64 , 100.8 ) .
State the final coordinates: ( 64 , 100.8 ) .
Explanation
Problem Analysis We are given two points, K ( 40 , 96 ) and J ( 80 , 104 ) , and we want to find 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 . We are also given the formulas to find the x and y coordinates of point P .
Calculate x-coordinate The formula for the x -coordinate of point P is given by: x = ( m + n m ) ( x 2 − x 1 ) + x 1 where x 1 = 40 , x 2 = 80 , and m + n m = 5 3 .
Substituting these values into the formula, we get: x = ( 5 3 ) ( 80 − 40 ) + 40 x = ( 5 3 ) ( 40 ) + 40 x = 24 + 40 x = 64 So, the x -coordinate of point P is 64.
Calculate y-coordinate The formula for the y -coordinate of point P is given by: y = ( m + n m ) ( y 2 − y 1 ) + y 1 where y 1 = 96 , y 2 = 104 , and m + n m = 5 3 .
Substituting these values into the formula, we get: y = ( 5 3 ) ( 104 − 96 ) + 96 y = ( 5 3 ) ( 8 ) + 96 y = 5 24 + 96 y = 4.8 + 96 y = 100.8 So, the y -coordinate of point P is 100.8.
Final Answer Therefore, the coordinates of point P are ( 64 , 100.8 ) .
Examples
In computer graphics, determining points along a line segment is crucial for rendering images and animations. For instance, if you want to draw a dashed line, you need to calculate the coordinates of the points where the dashes start and end. The formula used in this problem helps to find those intermediate points accurately, ensuring smooth and visually appealing graphics.
The coordinates of point P on the directed line segment from K to J, situated 5 3 the distance from K, are ( 64 , 100.8 ) . However, this answer does not match any of the provided options. Therefore, none of the options A through D are correct based on the calculations performed.
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