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.

Answer 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 )

Answer 2

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