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