Question

Determine whether the points are collinear.

 L(–2, 3), M(1, –3), N(5, 4)

Answer 1

L(–2, 3), M(1, –3), N(5, 4)

According to distance formula,

d(L, M) = `sqrt((x_2  –  x_1)^2 + (y_2  –  y_1)^2)`

d(L, M) = `sqrt([1  –  (–2)]^2 + (–3  –  3)^2)`

d(L, M) = `sqrt((1 + 2)^2 + (–3  –  3)^2)`

d(L, M) = `sqrt((3)^2 + (–6)^2)`

d(L, M) = `sqrt(9 + 36)`

d(L, M) = `sqrt(45)`

d(L, M) = `sqrt(9 × 5)`

∴ d(L, M) = `3sqrt(5)`                      ...(1)

d(M, N) = `sqrt((x_2  –  x_1)^2 + (y_2  –  y_1)^2)`

d(M, N) = `sqrt((5  –  1)^2 + [4  –  (– 3)]^2)`

d(M, N) = `sqrt((5  –  1)^2 + (4  +  3)^2)`

d(M, N) = `sqrt((4)^2 + (7)^2)`

d(M, N) = `sqrt(16 + 49)`

∴ d(M, N) = `sqrt(65)`                      ...(2)

d(L, N) = `sqrt((x_2  –  x_1)^2 + (y_2  –  y_1)^2)`

d(L, N) = `sqrt([5  –  (– 2)]^2 + (4  –  3)^2)`

d(L, N) = `sqrt((5  + 2)^2 + (4  –  3)^2)`

d(L, N) = `sqrt((7)^2 + (1)^2)`

d(L, N) = `sqrt(49 + 1)`

d(L, N) = `sqrt(50)`     

d(L, N) = `sqrt(25 × 2)`      

∴ d(L, N) = `5sqrt(2)`                      ...(3)  

From (1), (2), and (3),          

Sum of two sides is not equal to the third side.

Hence, the given points are not collinear.