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?
$x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$
$y=\left[\frac{m}{m+n}\right]\left(y_2-y_1\right)+y_1$
A. $(40,96)$
B. $(85,105)$
C. $(80,104)$
D. $(96,72)$
Answer (2)
Identify the coordinates of points K and J, and the fraction 5 3 representing the position of point P on the line segment.
Apply the formula x = ( m + n m ) ( x 2 − x 1 ) + x 1 to calculate the x-coordinate of point P, resulting in x = 67 .
Apply the formula y = ( m + n m ) ( y 2 − y 1 ) + y 1 to calculate the y-coordinate of point P, resulting in y = 101.4 .
State the coordinates of point P as ( 67 , 101.4 ) .
Explanation
Problem Analysis and Given Information We are given two points, K ( 40 , 96 ) and J ( 85 , 105 ) , 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 . We are also given the formulas for the x and y 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 K , ( x 2 , y 2 ) are the coordinates of point J , and the ratio m : n represents the fraction of the distance from K to J that point P is located.
Identify Given Values First, we identify the given values:
( x 1 , y 1 ) = ( 40 , 96 ) ( x 2 , y 2 ) = ( 85 , 105 ) m + n m = 5 3 . This implies that m = 3 and m + n = 5 , so n = 5 − m = 5 − 3 = 2 . Thus, m = 3 and n = 2 .
Calculate the x-coordinate Now, we substitute these values into the formula for the x -coordinate of point P :
x = ( 3 + 2 3 ) ( 85 − 40 ) + 40 x = ( 5 3 ) ( 45 ) + 40 x = 5 3 × 45 + 40 x = 3 × 9 + 40 x = 27 + 40 x = 67
Calculate the y-coordinate Next, we substitute the values into the formula for the y -coordinate of point P :
y = ( 3 + 2 3 ) ( 105 − 96 ) + 96 y = ( 5 3 ) ( 9 ) + 96 y = 5 3 × 9 + 96 y = 5 27 + 96 y = 5.4 + 96 y = 101.4
State the Coordinates of Point P Therefore, the coordinates of point P are ( 67 , 101.4 ) .
Examples
In city planning, determining the location of a new facility (like a bus stop or a park) along a street segment can be optimized using directed line segments. If you want to place the facility 5 3 of the way along a street from point A to point B, you can use the formula for finding a point on a directed line segment. This ensures equitable access and efficient use of space, enhancing community services.
By applying the formulas for finding the coordinates on the directed line segment from point K to point J, we calculated the coordinates of point P as (67, 101.4). Unfortunately, this result does not match any of the provided answer choices. Please verify the values of K and J for accuracy.
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