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)
Determine the ratio m + n m based on the given fraction of the line segment: 5 3 .
Calculate the x -coordinate of point P using the formula: x = 5 3 ( x 2 − x 1 ) + x 1 = 5 3 ( 80 − 40 ) + 40 = 64 .
Calculate the y -coordinate of point P using the formula: y = 5 3 ( y 2 − y 1 ) + y 1 = 5 3 ( 104 − 96 ) + 96 = 100.8 .
State the coordinates of point P: ( 64 , 100.8 ) .
Explanation
Understanding the Problem Let's break down this problem. We're given two points, K and J, and we want to find a point P that lies on the line segment connecting them. Point P is 5 3 of the way from K to J. We're also given formulas to calculate the x and y coordinates of P.
Identifying Given Information We are given the coordinates of point K as ( x 1 , y 1 ) = ( 40 , 96 ) and the coordinates of point J as ( x 2 , y 2 ) = ( 80 , 104 ) . We also know that P is 5 3 of the way from K to J, which means the ratio m : n is 3 : 2 . Therefore, m + n m = 3 + 2 3 = 5 3 .
Calculating the x-coordinate Now, let's use the given formula to find the x -coordinate of point P: x = ( m + n m ) ( x 2 − x 1 ) + x 1 Substitute the values: x = 5 3 ( 80 − 40 ) + 40 x = 5 3 ( 40 ) + 40 x = 24 + 40 x = 64
Calculating the y-coordinate Next, let's find the y -coordinate of point P using the given formula: y = ( m + n m ) ( y 2 − y 1 ) + y 1 Substitute the values: y = 5 3 ( 104 − 96 ) + 96 y = 5 3 ( 8 ) + 96 y = 5 24 + 96 y = 4.8 + 96 y = 100.8
Final Answer Therefore, the coordinates of point P are ( 64 , 100.8 ) .
Examples
In computer graphics, this concept is used to draw lines and curves. If you want to draw a line from point A to point B, you can calculate the points in between using the same formula. This is also useful in animation, where you need to move an object smoothly from one point to another.
The coordinates of point P, which is /5 of the way from K to J, are (64, 100.8). However, none of the answer choices provided matches these coordinates exactly. Thus, based on the calculated values, the proper coordinates point P is (64, 100.8).
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