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Question
Determine whether the given set of points are collinear or not
(7, −2), (5, 1), (3, 4)
Solution
To prove that three points are collinear, sum of the distance between two pairs of points is equal to the third pair of points.
Distance AB = `sqrt((5 - 7)^2 + (1 + 2)^2`
= `sqrt((-2)^2 + (3)^2`
= `sqrt(4 + 9)`
= `sqrt(13)`
BC = `sqrt((3 - 5)^2 + (4 - 1)^2`
= `sqrt((-2)^2 + (3)^2`
= `sqrt(4 + 9)`
= `sqrt(13)`
AC = `sqrt((3 - 7)^2 + (4 + 2)^2`
= `sqrt((-4)^2 + (6)^2`
= `sqrt(16 + 36)`
= `sqrt(52)`
= `sqrt(2 xx 2 xx 13)`
= `2sqrt(13)`
AB + BC = AC
`sqrt(13) + sqrt(13) = 2sqrt(13)`
⇒ `2sqrt(13) = 2sqrt(13)`
∴ The given three points are collinear.
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