7.2 Division of a Line Segment - SPM Additional Mathematics

7.2 Division of a Line Segment

(A) Midpoints of a Line Segment

Formula for the midpoint, M of A (x_l, 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 m and n.

Solution:

\begin{array}{l} B is the midpoint of A C \\ (m - 4, 3) = (\frac{- 1 + 5}{2}, \frac{n + 8}{2}) \\ (m - 4, 3) = (2, \frac{n + 8}{2}) \\ m - 4 = 2 and \frac{n + 8}{2} = 3 \\ m = 6 and n + 8 = 6 \\ n = - 2 \end{array}

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

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

Example 2:

The coordinate of R(2, –1) divide internally the line of AB with the ratio 3 : 2. If coordinate of A is (–1, 2), find the coordinate of B.

Solution:

\begin{array}{l} Let point B = (p, q) \\ (\frac{2 (- 1) + 3 p}{3 + 2}, \frac{2 (2) + 3 q}{3 + 2}) = (2, - 1) \\ (\frac{- 2 + 3 p}{5}, \frac{4 + 3 q}{5}) = (2, - 1) \\ Equating the x -coordinates, \\ \frac{- 2 + 3 p}{5} = 2 \\ - 2 + 3 p = 10 \\ 3 p = 12 \\ p = 4 \\ Equating the y -coordinates, \\ \frac{4 + 3 q}{5} = - 1 \\ 4 + 3 q = - 5 \\ 3 q = - 9 \\ q = - 3 \\ ∴ The coordinates of point B = (4, - 3) . \end{array}