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)
Set up the problem by identifying the coordinates of points K and J and the given ratio 5 3 .
Calculate the x-coordinate of point P using the formula x P = ( 5 3 ) ( 80 − 40 ) + 40 , which simplifies to x P = 64 .
Calculate the y-coordinate of point P using the formula y P = ( 5 3 ) ( 104 − 96 ) + 96 , which simplifies to y P = 100.8 .
State the final coordinates of point P as ( 64 , 100.8 ) .
Explanation
Problem Setup and Given Information 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 for the x- and y-coordinates of P:
x = ( m + n m ) ( x 2 − x 1 ) + x 1 y = ( m + n m ) ( y 2 − y 1 ) + y 1
Here, ( x 1 , y 1 ) = (40, 96) and ( x 2 , y 2 ) = (80, 104), and m + n m = 5 3 .
Calculating the Coordinates of Point P Now, we can plug in the given values into the formulas to find the coordinates of point P.
For the x-coordinate: x P = ( 5 3 ) ( 80 − 40 ) + 40 x P = ( 5 3 ) ( 40 ) + 40 x P = 5 3 × 40 + 40 x P = 5 120 + 40 x P = 24 + 40 x P = 64
For the y-coordinate: y P = ( 5 3 ) ( 104 − 96 ) + 96 y P = ( 5 3 ) ( 8 ) + 96 y P = 5 3 × 8 + 96 y P = 5 24 + 96 y P = 4.8 + 96 y P = 100.8
Final Answer Therefore, the coordinates of point P are (64, 100.8).
Examples
In computer graphics, when drawing a line from one point to another, you might want to place an object at a certain fraction along that line. For example, if you are drawing a road from city A to city B, you might want to place a gas station 5 3 of the way from city A to city B. This problem demonstrates how to calculate the exact coordinates of that gas station.
The coordinates of point P on the directed line segment from K(40, 96) to J(80, 104) calculated at 5 3 of the distance are (64, 100.8). This result shows how to apply the given formulas to find point P accurately. However, this answer does not match the multiple-choice options provided, so it is important to review the details again.
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