What are the $x$- and $y$-coordinates of point P on the directed line segment from $A$ to $B$ such that $P$ is $\frac{2}{3}$ length of the line segment from $A$ to $B$?
$\begin{array}{l}
x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1 \\
y=\left(\frac{m}{m+n}\right)\left(y_2-y_1\right)+y_1
\end{array}$
$(2,-1)$
$(4,-3)$
$(-1,2)$
$(3,-2)$
Answer (1)
Determine the ratio m : n as 2 : 1 since P is 3 2 of the way from A to B .
Use the section formula x = ( m + n m ) ( x 2 − x 1 ) + x 1 to find the x -coordinate of P .
Use the section formula y = ( m + n m ) ( y 2 − y 1 ) + y 1 to find the y -coordinate of P .
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 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, ( x 1 , y 1 ) = ( 2 , − 1 ) , ( x 2 , y 2 ) = ( 4 , − 3 ) , and m : n = 2 : 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 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 y ≈ − 2.33
Final Answer Therefore, the coordinates of point P are ( 3 10 , − 3 7 ) or approximately ( 3.33 , − 2.33 ) .
Examples
In computer graphics, when drawing a line between two points, you might want to find a point that is a certain fraction of the way along that line. This is useful for creating animations or drawing dashed lines. The section formula helps calculate the coordinates of that intermediate point.
