Using distance formula show that distance points are collinear $(1, - 1), (5, 2), (9, 5)$ .

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Solution

A $(1, - 1)$ ,B $(5, 2)$ ,C $(9, 5)$
AB $= \sqrt{{(5 - 1)}^{2} + {(2 - (- 1))}^{2}} = \sqrt{4^{2} + 3^{2}} = \sqrt{25} = 5$
BC $= \sqrt{{(9 - 5)}^{2} + {(5 - 2)}^{2}} = \sqrt{4^{2} + 3^{2}} = \sqrt{25} = 5$
CA $= \sqrt{{(9 - 1)}^{2} + (5 - {(- 1)}^{2})} = \sqrt{8^{2} + 6^{2}} = \sqrt{100} = 10$
$∴$ as $A B + B C = C A$
A,B,C are collinear.

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