Question

Show that the points (1, 3), (2, 1) and `(1/2, 4)` are collinear, by using any other method

Answer 1

Let the given points be A(1, 3), B(2, 1) and `"C"(1/2, 4)`

Distance method:

A(1, 3), B(2, 1), `"C"(1/2, 4)`

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

AB = `sqrt(1 + 4)`

= `sqrt(5)`

BC = `sqrt((1/2 + 2)^2 + (4 - 1)^2`

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

= `sqrt(9/4 + 9)`

= `sqrt((9 + 36)/4`

= `sqrt(45/4)`

= `sqrt((9 xx 5)/4`

BC = `3/2 sqrt(5)`

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

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

= `sqrt(1/4 + 1)`

= `sqrt((1 + 4)/4`

=`sqrt(5/4)`

AC = `1/2 sqrt(5)`

AB + AC = `sqrt(5) + 1/2 sqrt(5)`

= `sqrt(5) (1 + 1/2)`

BA + AC = `3/2 sqrt(5)`

= BC

Thus BA + AC = BC

∴ The points A, B, C are collinear.