What are the $x$- and $y$-coordinates of point P on the directed line segment from $A$ to $B$ such that $P$ is $\frac{2}{3}$ the length of the line segment from $A$ to $B$?
$x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$
$y=\left(\frac{m}{m+n}\right)\left(y_2-y_1\right)+y_1$
(2,-1)
(4,-3)
(-1,2)
(3,-2)
Answer (2)
Determine the ratio m : n as 2 : 1 since P is 3 2 of the way from A to B .
Apply the section formula for the x -coordinate: x = ( 2 + 1 2 ) ( 4 − 2 ) + 2 = 3 10 .
Apply the section formula for the y -coordinate: y = ( 2 + 1 2 ) ( − 3 − ( − 1 )) + ( − 1 ) = − 3 7 .
State the coordinates of point P : ( 3 10 , − 3 7 ) .
Explanation
Problem Analysis We are given two points, A ( 2 , − 1 ) and B ( 4 , − 3 ) , and we want to find the coordinates of point P on the directed line segment from A to B such that P is 3 2 of the length of the line segment from A to B . This means that P divides the line segment A B in the ratio 2 : 1 . We can use the section formula to find the coordinates of point P .
Section Formula The section formula is given by: x = ( m + n m ) ( x 2 − x 1 ) + x 1 y = ( m + n m ) ( y 2 − y 1 ) + y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of points A and B respectively, and m : n is the ratio in which P divides the line segment A B . In our case, A ( 2 , − 1 ) , B ( 4 , − 3 ) , and the ratio is 2 : 1 , so m = 2 and n = 1 .
Calculate x-coordinate Now, we can substitute the given values into the section formula to find the x -coordinate of point P :
x = ( 2 + 1 2 ) ( 4 − 2 ) + 2 x = ( 3 2 ) ( 2 ) + 2 x = 3 4 + 2 x = 3 4 + 3 6 x = 3 10 x = 3 3 1 x ≈ 3.33
Calculate y-coordinate Next, we substitute the given values into the section formula to find the y -coordinate of point P :
y = ( 2 + 1 2 ) ( − 3 − ( − 1 )) + ( − 1 ) y = ( 3 2 ) ( − 3 + 1 ) − 1 y = ( 3 2 ) ( − 2 ) − 1 y = − 3 4 − 1 y = − 3 4 − 3 3 y = − 3 7 y = − 2 3 1 y ≈ − 2.33
Final Answer Therefore, the coordinates of point P are ( 3 10 , − 3 7 ) or approximately ( 3.33 , − 2.33 ) .
Examples
In architecture, when designing a building facade, you might want to divide a vertical line segment representing the height of the facade into specific ratios to place windows or decorative elements. If the total height is the line segment AB, and you want to place a window at a point P that is 2/3 of the way up the facade, you can use the section formula to calculate the exact coordinates (height) where the window should be placed to achieve the desired aesthetic proportion.
The coordinates of point P, which is 3 2 of the way from A(2, -1) to B(4, -3), are ( 3 10 , − 3 7 ) .
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