Let us prove that the midpoint of line segment joining two points (2, 1) and (6, 5) lie on the line joining two points (–4, –5) and (9, 8).

Le the points be M = (2, 1) and N = (6, 5)

Midpoint of line segment joining two points is given by

Where (M_x, M_y) are x and y coordinates of point M and (N_x, N_y) are x and y coordinates of point N

⇒ Midpoint =

⇒ Midpoint = (4, 3)

Let this point be A (4, 3)

Let the points be B and C as follows

B = (–4, –5) and C = (9, 8)

Now, we have to prove that the midpoint i.e. A lies on line joining points B and C

If we prove that area of triangle formed by joining points A, B and C is 0 then A, B and C will be collinear and thus we can say that point A lies on line joining points B and C

Thus the 3 vertices of triangle here are

A = (x₁, y₁) = (4, 3)

B = (x₂, y₂) = (-4, -5)

C = (x₃, y₃) = (9, 8)

So we have to prove that area(ΔACB) = 0

Area of triangle is given by formula

Area = × [x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂)]

Where (x₁, y₁), (x₂, y₂) and (x₃, y₃) are vertices of triangle

Substituting values

⇒ Area = × [4(-5 – 8) + (-4)(8 – 3) + 9(3 – (-5))]

⇒ Area = × [-52 + (-20) + 72]

⇒ Area = × [- 72 + 72]

⇒ Area = 0

Hence midpoint of (2, 1) and (6, 5) lie on the line joining two points (–4, –5) and (9, 8)