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)
Point P divides the directed line segment from A to B in the ratio 2 : 1 .
Apply the section formula to find the x-coordinate: x = ( 3 2 ) ( 4 − 2 ) + 2 = 3 10 .
Apply the section formula to find the y-coordinate: y = ( 3 2 ) ( − 3 − ( − 1 )) + ( − 1 ) = − 3 7 .
The coordinates of point P are ( 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 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 A ( x 1 , y 1 ) and B ( x 2 , y 2 ) are the given points, and m : n is the ratio in which P divides the line segment A B . In our case, A ( 2 , − 1 ) , B ( 4 , − 3 ) , 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
Calculate y-coordinate Next, we can 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
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, you might need to divide a wall into specific ratios to place windows or doors proportionally. The section formula helps determine the exact coordinates for these placements, ensuring aesthetic balance and structural integrity.
The coordinates of point P, located 2/3 of the way along the directed line segment from point A(2, -1) to point B(4, -3), are \left(\frac{10}{3}, -\frac{7}{3}\right). This means P is approximately (3.33, -2.33).
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