6.2 Division of a Line Segment

6.2 Division of a Line Segment

(A) Midpoints of a Line Segment

Formula for the midpoint, Mof A(xl, y1) and B(x2, y2) is


Example 1:
Given B (m – 4, 3) is the midpoint of the straight line joining A(–1, n) and C (5, 8). Find the values of mand n.

Solution:
B  is the midpoint of  A C ( m 4 ,   3 ) = ( 1 + 5 2 ,   n + 8 2 ) ( m 4 ,   3 ) = ( 2 ,   n + 8 2 ) m 4 = 2          and         n + 8 2 = 3 m = 6                 and           n + 8 = 6                                                n = 2


(B) Point that Internally Divides a Line Segment in the 
Ratio m : n

Formula for the point Pthat lies on AB such that AP : PB = m : n is


Example 2:
The coordinate of R(2, –1) divide internally the line of ABwith the ratio 3 : 2. If coordinate of Ais (–1, 2), find the coordinate of B.

Solution:


Let point  B = ( p ,   q ) ( 2 ( 1 ) + 3 p 3 + 2 ,   2 ( 2 ) + 3 q 3 + 2 ) = ( 2 , 1 ) ( 2 + 3 p 5 ,   4 + 3 q 5 ) = ( 2 , 1 ) Equating the  x -coordinates, 2 + 3 p 5 = 2 2 + 3 p = 10 3 p = 12 p = 4 Equating the  y -coordinates, 4 + 3 q 5 = 1 4 + 3 q = 5 3 q = 9 q = 3  The coordinates of point  B = ( 4 , 3 ) .

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