What are the $x$- and $y$-coordinates of point P on the directed line segment from K to J such that P is $\frac{3}{5}$ the length of the line segment from K to J?
$\begin{array}{l}<
x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1 \\
y=\left(\frac{m}{m+n}\right)\left(y_2-y_1\right)+y_1
\end{array}$
A. $(40,96)$
B. $(85,105)$
C. $(80,104)$
D. $(96,72)$
Answer (2)
Define the coordinates of points K and J and the given ratio.
Apply the section formula to find the x-coordinate of point P: x P = ( 5 3 ) ( 80 − 40 ) + 40 = 64 .
Apply the section formula to find the y-coordinate of point P: y P = ( 5 3 ) ( 104 − 96 ) + 96 = 100.8 .
State the coordinates of point P: ( 64 , 100.8 ) .
Explanation
Problem Analysis and Setup We are given two points, K(40, 96) and J(80, 104), and we want to find the coordinates of point P on the directed line segment from K to J such that P is 5 3 the length of the line segment from K to J. This means that the ratio of KP to PJ is 3:2. We can use the section formula to find the coordinates of point P.
Stating the 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, y) are the coordinates of point P, ( x 1 , y 1 ) are the coordinates of point K, ( x 2 , y 2 ) are the coordinates of point J, and m:n is the ratio in which P divides the line segment KJ. In our case, K = (40, 96), J = (80, 104), and m:n = 3:2.
Calculating the x-coordinate of P Now, we can plug in the values into the section formula to find the x-coordinate of point P:
x P = ( 3 + 2 3 ) ( 80 − 40 ) + 40
x P = ( 5 3 ) ( 40 ) + 40
x P = 24 + 40
x P = 64
Calculating the y-coordinate of P Next, we can plug in the values into the section formula to find the y-coordinate of point P:
y P = ( 3 + 2 3 ) ( 104 − 96 ) + 96
y P = ( 5 3 ) ( 8 ) + 96
y P = 5 24 + 96
y P = 4.8 + 96
y P = 100.8
Final Answer Therefore, the coordinates of point P are (64, 100.8).
Examples
In architecture, when designing a building facade, you might want to divide a vertical line segment representing the height of a window into specific ratios to place decorative elements. Using the section formula, you can accurately calculate the coordinates (positions) of these elements, ensuring they are proportionally spaced and aesthetically pleasing. For instance, if a window's height spans from (0, 0) to (0, 5) and you want to place a decorative stripe 5 2 of the way up, the section formula helps you find the exact y-coordinate for the stripe's placement.
Point P on the line segment from K(40, 96) to J(80, 104) is calculated to be at the coordinates (64, 100.8). To find these coordinates, we used the section formula with the ratio of 3:2. This result does not match the given multiple-choice options.
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