Question

The mid-point of the line segment AB shown in adjoining figure is (5, −4). Find the co-ordinates of A and B.

Answer 1

If A = (x₁, y₁) and B = (x₂, y₂),

The midpoint M is:

M = `((x_1 + x_2)/2, (y_1 + y_2)/2)`

Let’s, apply to M = (5, −4)

⇒ `(x_1 + x_2)/2` = 5

x₁ + x₂ = 10

⇒ `(y_1 + y_2)/2` = −4

y₁ + y₂ = −8

Here, there are infinitely many pairs (A, B) satisfying those two equations. In general, if A = (x₁, y₁), then B = (10 − x₁, −8 − y₁).

For examples,

• If A = (10, 0),
  then B = (0, −8)
• If A = (7, −1),
  then B = (3, −7)
• If A = (5 + k, −4 + m),
  then B = (5 − k, −4 − m) for any numbers k, m