What are the $x$- and $y$-coordinates of point $E$, which partitions the directed line segment from $A$ to $B$ into a ratio of $1:2$?
[tex]x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1[/tex]
[tex]v=\left(\frac{m}{m+n}\right)\left(v_2-v_1\right)+v_1[/tex]
Answer (2)
Use the section formula to find the coordinates of point E that divides the line segment from A to B in the ratio 1 : 2 .
Substitute the given values m = 1 , n = 2 , x 1 = 0 , y 1 = 1 , x 2 = − 2 , and y 2 = 5 into the section formula.
Calculate the x -coordinate: x = ( 3 1 ) ( − 2 − 0 ) + 0 = − 3 2 .
Calculate the y -coordinate: y = ( 3 1 ) ( 5 − 1 ) + 1 = 3 7 .
State the coordinates of point E : ( − 3 2 , 3 7 ) .
Explanation
Problem Analysis and Setup We are given the coordinates of points A ( 0 , 1 ) and B ( − 2 , 5 ) . We want to find the coordinates of point E that partitions the directed line segment from A to B in the ratio 1 : 2 . The section formulas for the x and y coordinates are provided:
x = ( m + n m ) ( x 2 − x 1 ) + x 1 y = ( m + n m ) ( y 2 − y 1 ) + y 1
where m : n is the ratio in which the line segment is divided, ( x 1 , y 1 ) are the coordinates of point A , and ( x 2 , y 2 ) are the coordinates of point B .
Identify Given Values We are given m = 1 , n = 2 , x 1 = 0 , y 1 = 1 , x 2 = − 2 , and y 2 = 5 . We will substitute these values into the section formulas to find the coordinates of point E .
Calculate x-coordinate First, let's find the x -coordinate of point E :
x = ( 1 + 2 1 ) ( − 2 − 0 ) + 0 x = ( 3 1 ) ( − 2 ) + 0 x = − 3 2
Calculate y-coordinate Now, let's find the y -coordinate of point E :
y = ( 1 + 2 1 ) ( 5 − 1 ) + 1 y = ( 3 1 ) ( 4 ) + 1 y = 3 4 + 1 y = 3 4 + 3 3 y = 3 7
State the Coordinates of Point E Therefore, the coordinates of point E are ( − 3 2 , 3 7 ) .
Examples
Imagine you're designing a video game and need to place a treasure chest at a specific point between two landmarks on the game map. If you want the treasure chest to be one-third of the way from Landmark A to Landmark B, you can use the section formula to calculate the exact coordinates where the treasure chest should be placed. This ensures the treasure is placed proportionally between the two landmarks, creating a balanced and intuitive gaming experience for the player.
The coordinates of point E, which partitions the directed line segment from point A to point B in a ratio of 1:2, are − 3 2 , 3 7 . This result is derived using the section formula for coordinates. Thus, point E is located at ( − 3 2 , 3 7 ) .
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