Determine whether the points are collinear.A(1, -3), B(2, -5), C(-4, 7) – Geometry | Husharmulga.com

By distance formula,

A(1, –3), B(2, –5), C(–4, 7)

\[ =d(A, B) \sqrt{((2 - 1)^2 + [-5 - (-3)]^2)} \]

\[ ∴ d(A, B)= \sqrt{((1)^2 + (-5 + 3)^2)}\]

\[∴ d(A, B)= \sqrt{(1)^2 + (-2)^2)}\]

\[∴ d(A, B) = \sqrt{(1+ 4)}\]

\[ ∴ d(A, B) = \sqrt{(5)}           ...(1)\]

\[d(B, C) = \sqrt((- 4 - 2)^2 + [7 - (-5)]^2) \]

\[ ∴ d(B, C) = \sqrt((-6)^2 + [7 + 5]^2) \]

\[∴ d(B, C) = \sqrt((-6)^2 + (12)^2) \]

\[∴ d(B, C) = \sqrt(36 + 144) \]

\[∴ d(B, C) = \sqrt(180) \]

\[ ∴ d(B, C) = \sqrt(36 × 5) \]

\[ ∴ d(B, C) =6 \sqrt(5)           ...(2) \]

\[d(A, C) = \sqrt([-4 - 1]^2 + [7 - (-3)]^2) \]

\[{ ∴ d(A, C) = \sqrt((-4 - 1)^2 + (7 + 3)^2)} \]

\[∴ d(A, C) = \sqrt((-5)^2 + (10)^2) \]

\[∴ d(A, C) = \sqrt(25 + 100) \]

\[∴ d(A, C) = \sqrt(125) \]

\[∴ d(A, C) = \sqrt(25 × 5) \]

\[∴ d(A, C) = 5 \sqrt{(5)}`           ...(3)\]

Adding (1) and (3)

∴ d(A, B) + d(A, C) = d(B, C)

∴ √5 + 5√5 = 6√5               ...(4)

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

∴ Points A,B and C are collinear.

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