What are the $x$- and $y$-coordinates of point $E$, which partitions the directed line segment from J to K into a ratio of 1:4?
$x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$
$V=\left(\frac{m}{m+n}\right)\left(y_2-y_1\right)+y_1$
A. $(-13,-3)$
B. $(-7,-1)$
C. $(-5,0)$
D. $(17,11)$
Answer (2)
Define the coordinates of points J and K, and the ratio m:n.
Substitute the given values into the section formula for the x-coordinate and calculate x.
Substitute the given values into the section formula for the y-coordinate and calculate y.
State the coordinates of point E as ( − 7 , − 0.2 ) .
Explanation
Problem Analysis We are given two points, J(-13, -3) and K(17, 11), and we want to find the coordinates of point E that partitions the directed line segment from J to K in the ratio 1:4. We are also given the formulas for calculating the x- and y-coordinates of point E.
Define the variables Let J be ( x 1 , y 1 ) = ( − 13 , − 3 ) and K be ( x 2 , y 2 ) = ( 17 , 11 ) . The ratio is given as m : n = 1 : 4 , so m = 1 and n = 4 .
Calculate x-coordinate Now, we substitute the values into the formula for the x-coordinate: x = ( m + n m ) ( x 2 − x 1 ) + x 1 x = ( 1 + 4 1 ) ( 17 − ( − 13 )) + ( − 13 ) x = ( 5 1 ) ( 17 + 13 ) − 13 x = ( 5 1 ) ( 30 ) − 13 x = 6 − 13 x = − 7
Calculate y-coordinate Next, we substitute the values into the formula for the y-coordinate: y = ( m + n m ) ( y 2 − y 1 ) + y 1 y = ( 1 + 4 1 ) ( 11 − ( − 3 )) + ( − 3 ) y = ( 5 1 ) ( 11 + 3 ) − 3 y = ( 5 1 ) ( 14 ) − 3 y = 5 14 − 3 y = 5 14 − 5 15 y = − 5 1 y = − 0.2
Final Answer Therefore, the coordinates of point E are (-7, -0.2).
Examples
In architecture, when designing a building facade, you might want to divide a long beam into specific ratios to place support columns. Using the section formula, you can accurately determine the coordinates for these support points, ensuring structural integrity and aesthetic appeal.
The coordinates of point E, which partitions the directed line segment from J to K in a ratio of 1:4, are (-7, -0.2). This corresponds to option B in the provided choices.
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