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]y=\left(\frac{m}{m+n}\right)\left(v_2-v_1\right)+v_1[/tex]
Answer (2)
Use the section formula to find the x-coordinate of point E: x = ( 1 + 2 1 ) ( − 1 − 0 ) + 0 = − 3 1 .
Use the section formula to find the y-coordinate of point E: y = ( 1 + 2 1 ) ( 3 − 1 ) + 1 = 3 5 .
The coordinates of point E are ( − 3 1 , 3 5 ) .
The final answer is ( − 3 1 , 3 5 ) .
Explanation
Problem Analysis We are given two points, A ( 0 , 1 ) and B ( − 1 , 3 ) , and we want to find the coordinates of point E that partitions the directed line segment from A to B in the ratio 1 : 2 . We are also given the formulas for the x and y coordinates of the point E .
Calculate the x-coordinate The section formula for the x -coordinate is given by: x = ( m + n m ) ( x 2 − x 1 ) + x 1 where m : n is the ratio, ( x 1 , y 1 ) are the coordinates of point A , and ( x 2 , y 2 ) are the coordinates of point B .
In our case, m = 1 , n = 2 , x 1 = 0 , and x 2 = − 1 . Substituting these values into the formula, we get: x = ( 1 + 2 1 ) ( − 1 − 0 ) + 0 = 3 1 ( − 1 ) + 0 = − 3 1
Calculate the y-coordinate The section formula for the y -coordinate is given by: y = ( m + n m ) ( y 2 − y 1 ) + y 1 where m = 1 , n = 2 , y 1 = 1 , and y 2 = 3 . Substituting these values into the formula, we get: y = ( 1 + 2 1 ) ( 3 − 1 ) + 1 = 3 1 ( 2 ) + 1 = 3 2 + 1 = 3 5
State the coordinates of point E Therefore, the coordinates of point E are ( − 3 1 , 3 5 ) .
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 chest should be placed. This ensures the treasure is located precisely where you intend it to be, creating a consistent and fair gaming experience for players.
;
