Question

Determine whether the points are collinear.

A(1, −3), B(2, −5), C(−4, 7)

Verify whether points A(1, −3), B(2, −5) and C(−4, 7) are collinear or not.

Answer 1

Given: A(1, −3), B(2, −5), C(−4, 7)

Let,

A(1, −3) = A(x₁, y₁)

B(2, −5) = B(x₂, y₂)

C(−4, 7) = C(x₃, y₃)

By the distance formula,

d(A, B) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

= `sqrt((2 - 1)^2 + [-5 - (-3)]^2)`

= `sqrt((1)^2 + (-5 + 3)^2)`

= `sqrt((1)^2 + (-2)^2)`

= `sqrt(1+ 4)`

= `sqrt(5)`    ...(1)

d(B, C) = `sqrt((x_3 - x_2)^2 + (y_3 - y_2)^2)`

= `sqrt((- 4 - 2)^2 + [7 - (-5)]^2)`

= `sqrt((-6)^2 + [7 + 5]^2)`

= `sqrt((-6)^2 + (12)^2)`

= `sqrt(36 + 144)`

= `sqrt(180)`

= `sqrt(36 xx 5)`

= `6sqrt(5)`    ...(2)

d(A, C) = `sqrt((x_3 - x_1)^2 + (y_3 - y_1)^2)`

= `sqrt((-4 - 1)^2 + [7 - (-3)]^2)`

= `sqrt((-4 - 1)^2 + (7 + 3)^2)`

= `sqrt((-5)^2 + (10)^2)`

= `sqrt(25 + 100)`

= `sqrt(125)`

= `sqrt(25 × 5)`

= `5sqrt(5)`     ...(3)

Adding (1) and (3)

∴ d(A, B) + d(A, C) = d(B, C)

∴ `sqrt5 + 5sqrt5 = 6sqrt5`    ...(4)

∴ d(A, B) + d(A, C) = d(B, C)  ...[From (2) and (4)]

∴ Points A(1, −3), B(2, −5) and C(−4, 7) are collinear.

Hence proved.