What are the $x$- and $y$-coordinates of point $E$, which partitions the directed line segment from J to K into a ratio of 1:4?
$\begin{array}{l}
x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1 \\
v=\left(\frac{m}{m+n}\right)\left(v_2-v_1\right)+v_1
\end{array}$
Answer (1)
Assign coordinates to points J and K: J ( − 13 , − 3 ) , K ( − 7 , − 1 ) .
Apply the section formula for the x-coordinate: x = ( 1 + 4 1 ) ( − 7 − ( − 13 )) + ( − 13 ) = − 11.8 .
Apply the section formula for the y-coordinate: y = ( 1 + 4 1 ) ( − 1 − ( − 3 )) + ( − 3 ) = − 2.6 .
State the coordinates of point E: ( − 11.8 , − 2.6 ) .
Explanation
Problem Analysis We are given two points, J and K, and we want to find the coordinates of point E that divides the directed line segment from J to K in the ratio 1:4. We are also given the formulas to calculate the x- and y-coordinates of point E.
Assign Values Let's denote the coordinates of point J as ( x 1 , y 1 ) = ( − 13 , − 3 ) and the coordinates of point K as ( x 2 , y 2 ) = ( − 7 , − 1 ) . The ratio is given as m : n = 1 : 4 , so m = 1 and n = 4 .
Calculate x-coordinate Now, we will use the section formula to find the x-coordinate of point E: x = ( m + n m ) ( x 2 − x 1 ) + x 1 Substituting the given values: x = ( 1 + 4 1 ) ( − 7 − ( − 13 )) + ( − 13 ) x = ( 5 1 ) ( − 7 + 13 ) − 13 x = ( 5 1 ) ( 6 ) − 13 x = 5 6 − 13 x = 1.2 − 13 x = − 11.8
Calculate y-coordinate Next, we will use the section formula to find the y-coordinate of point E: y = ( m + n m ) ( y 2 − y 1 ) + y 1 Substituting the given values: y = ( 1 + 4 1 ) ( − 1 − ( − 3 )) + ( − 3 ) y = ( 5 1 ) ( − 1 + 3 ) − 3 y = ( 5 1 ) ( 2 ) − 3 y = 5 2 − 3 y = 0.4 − 3 y = − 2.6
Final Answer Therefore, the coordinates of point E are ( − 11.8 , − 2.6 ) .
Examples
In computer graphics, when drawing a line between two points, you might want to place an object at a certain fraction along that line. This problem demonstrates how to calculate the exact coordinates for placing that object, ensuring it's positioned correctly relative to the start and end points.
