Given the points A(-8, 6) and B(16, 12), determine the coordinates of point P on directed line segment that partitions in the ratio 3:1.
The formula used for internally division of a line segment in the ratio a:b is
x coordinate = ( a ∙ x2 + b ∙ x1 ) / ( a + b )
y coordinate = ( a ∙ y2 + b ∙ y1 ) / ( a + b )
In this case:
a = 3 , b = 1 , x1 = - 8 , y1 = 6 , x2 = 16 , y2 = 12
x coordinate = [ 3 ∙ 16 + 1 ∙ ( - 8 ) ] / ( 3 + 1 ) = ( 48 - 8 ) / 4 = 40 / 4 = 10
y coordinate = ( 3 ∙ 12 + 1 ∙ 6 ) / ( 3 + 1 ) = ( 36 + 6 ) / 4 = 42 / 4 = 10.5
The coordinates of point P is ( 10 , 10.5 )
P is 3/4 of the way from A to B. So,
P = A + 3/4 (B-A)
so, for the x-coordinate,
Px = -8 + 3/4 (16+8) = -8 + 3/4 * 24 = -8 + 18 = 10
Do Py the same way
To determine the coordinates of point P on a directed line segment that partitions in the ratio 3:1, we can use the concept of linear interpolation.
Step 1: Find the difference in x-coordinates and y-coordinates of points A and B.
Let's call the difference in x-coordinates Δx = xB - xA = 16 - (-8) = 24.
And the difference in y-coordinates Δy = yB - yA = 12 - 6 = 6.
Step 2: Determine the coordinates of point P by interpolating between points A and B.
Given that the line segment is partitioned in a ratio of 3:1, we need to divide Δx and Δy by 4 (3 + 1) to get the segment lengths in the ratio. So, we have:
Segment length in x-direction = Δx/4 = 24/4 = 6.
Segment length in y-direction = Δy/4 = 6/4 = 1.5.
To find the coordinates of point P, we start from point A and move 3 units in the x-direction and 1.5 units in the y-direction:
Px = Ax + 3*(segment length in x-direction) = -8 + 3*6 = 10.
Py = Ay + 1.5*(segment length in y-direction) = 6 + 1.5*1.5 = 8.25.
Therefore, the coordinates of point P are P(10, 8.25).