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 $B$?
$x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$
$y=\left(\frac{m}{m+n}\right)\left(y_2-y_1\right)+y_1$
A. (2,-1)
B. (4,-3)
C. (-1,2)
D. (3,-2)
Answer (2)
Determine the ratio m : n as 2 : 1 since P is 3 2 of the way from A to B .
Use the section formula to calculate the x -coordinate of P : x = ( 2 + 1 2 ) ( 4 − 2 ) + 2 = 3 10 .
Use the section formula to calculate the y -coordinate of P : y = ( 2 + 1 2 ) ( − 3 − ( − 1 )) + ( − 1 ) = − 3 7 .
State the coordinates of point P : ( 3 10 , − 3 7 ) .
Explanation
Problem Analysis and Given Information 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 segment A B in the ratio 2 : 1 . We are also given the formulas for finding the coordinates of point P :
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 ) are the coordinates of point A , ( x 2 , y 2 ) are the coordinates of point B , and m : n is the ratio in which P divides the segment A B .
Calculating the x-coordinate In this case, we have A ( 2 , − 1 ) , B ( 4 , − 3 ) , m = 2 , and n = 1 . We can now substitute these values into the formulas to find the coordinates of point P .
For the x -coordinate:
x = ( 2 + 1 2 ) ( 4 − 2 ) + 2 x = ( 3 2 ) ( 2 ) + 2 x = 3 4 + 2 x = 3 4 + 3 6 x = 3 10
Calculating the y-coordinate For the y -coordinate:
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
Final Answer Therefore, the coordinates of point P are ( 3 10 , − 3 7 ) .
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. For instance, if you want a window to be positioned two-thirds of the way up the facade, you can use the section formula to calculate the exact coordinates (height) where the window should be placed, ensuring a balanced and aesthetically pleasing design.
To find the coordinates of point P dividing the segment from A(2, -1) to B(4, -3) in a 2:1 ratio, we used the section formula. The calculated coordinates of point P are ( 3 10 , − 3 7 ) , which do not match any provided multiple-choice answers. This indicates a potential issue with the given options, as mathematically our results are valid.
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