Using the distance formula, show that the points (1, −1), (5, 2) and (9, 5) are collinear.

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Solution

Let A(1, −1), B (5, 2) and C(9, 5) be the given points. Then:
$A B = \sqrt{{(5 - 1)}^{2} + {(2 + 1)}^{2}} = \sqrt{{(4)}^{2} + {(3)}^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5 units B C = \sqrt{{(9 - 5)}^{2} + {(5 - 2)}^{2}} = \sqrt{{(4)}^{2} + {(3)}^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5 units A C = \sqrt{{(9 - 1)}^{2} + {(5 + 1)}^{2}} = \sqrt{{(8)}^{2} + {(6)}^{2}} = \sqrt{64 + 36} = \sqrt{100} = 10 units A B + B C = (5 + 5) = 10 units = A C$

Hence, the given points are collinear.

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Show that the points (1, – 1), (5, 2) and (9, 5) are collinear.

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