In the coordinate plane, the directed line segment from $K$ to $N$ has endpoints at $K(-6,-2)$ and $N(8,3)$. Point $L$ partitions the directed line segment from K to N in a ratio of $1: 2$. Point M partitions the directed line segment from L to N in a ratio of $3: 1$. What are the coordinates of point M? Round to the nearest tenth, if necessary.
A. ( $-1.3,-0.3$ )
B. (2.5, 1)
C. (5.7, 2.2)
D. (7, 2.5)
Answer (1)
Find the coordinates of point L using the section formula with the given ratio 1:2: L = ( 1 + 2 1 ( 8 ) + 2 ( − 6 ) , 1 + 2 1 ( 3 ) + 2 ( − 2 ) ) = ( − 3 4 , − 3 1 ) .
Find the coordinates of point M using the section formula with the given ratio 3:1 and the coordinates of L and N: M = ( 3 + 1 3 ( 8 ) + 1 ( − 3 4 ) , 3 + 1 3 ( 3 ) + 1 ( − 3 1 ) ) = ( 3 17 , 6 13 ) .
Approximate the coordinates of point M to two decimal places: M ≈ ( 5.67 , 2.17 ) .
Round the coordinates of point M to the nearest tenth: ( 5.7 , 2.2 ) .
Explanation
Problem Analysis We are given the coordinates of points K ( − 6 , − 2 ) and N ( 8 , 3 ) . Point L divides the segment K N in the ratio 1 : 2 , and point M divides the segment L N in the ratio 3 : 1 . We need to find the coordinates of point M .
Finding Coordinates of Point L First, we find the coordinates of point L using the section formula. The section formula for a point dividing a line segment with endpoints ( x 1 , y 1 ) and ( x 2 , y 2 ) in the ratio m : n is given by:
( m + n m x 2 + n x 1 , m + n m y 2 + n y 1 )
For point L , we have K ( − 6 , − 2 ) and N ( 8 , 3 ) with the ratio 1 : 2 . So, m = 1 , n = 2 , x 1 = − 6 , y 1 = − 2 , x 2 = 8 , and y 2 = 3 . Plugging these values into the section formula, we get:
L = ( 1 + 2 1 ( 8 ) + 2 ( − 6 ) , 1 + 2 1 ( 3 ) + 2 ( − 2 ) ) = ( 3 8 − 12 , 3 3 − 4 ) = ( 3 − 4 , 3 − 1 )
So, the coordinates of point L are ( − 3 4 , − 3 1 ) ≈ ( − 1.33 , − 0.33 ) .
Finding Coordinates of Point M Next, we find the coordinates of point M using the section formula again. This time, point M divides the segment L N in the ratio 3 : 1 . We have L ( − 3 4 , − 3 1 ) and N ( 8 , 3 ) with the ratio 3 : 1 . So, m = 3 , n = 1 , x 1 = − 3 4 , y 1 = − 3 1 , x 2 = 8 , and y 2 = 3 . Plugging these values into the section formula, we get:
M = ( 3 + 1 3 ( 8 ) + 1 ( − 3 4 ) , 3 + 1 3 ( 3 ) + 1 ( − 3 1 ) ) = ( 4 24 − 3 4 , 4 9 − 3 1 ) = ( 4 3 72 − 4 , 4 3 27 − 1 ) = ( 4 3 68 , 4 3 26 ) = ( 12 68 , 12 26 ) = ( 3 17 , 6 13 )
So, the coordinates of point M are ( 3 17 , 6 13 ) ≈ ( 5.67 , 2.17 ) .
Final Answer Rounding the coordinates of point M to the nearest tenth, we get ( 5.7 , 2.2 ) .
Conclusion Therefore, the coordinates of point M are approximately ( 5.7 , 2.2 ) .
Examples
In computer graphics, determining the coordinates of points that divide a line segment in a given ratio is crucial for rendering images and creating animations. For instance, when drawing a line or curve, the algorithm might need to calculate intermediate points to ensure a smooth appearance. The section formula is used to find these intermediate points, allowing for precise control over the shape and position of graphical elements.
