Question

prove that the points (4,-5),(1,1)and (-2,7) are collinier by distance formula ​

Answer 1

Solution :-

Let the points be A ( 4 , -5 ), B ( 1 , 1 ) and C ( -2 , 7 ).

$\underline{\boxed{\bf Distance \ formula = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} }}$

$\bullet \sf \ AB = \sqrt{(1-4)^2+(1-(-5))^2}$

        $= \sf \sqrt{(-3)^2+6^2} \\\\= \sqrt{9+36} \\\\= \sqrt{45} \\\\= \sqrt{9 \times 5 } \\\\= 3 \sqrt{5} \ units$

$\bullet \sf \ BC = \sqrt{(-2-1)^2+(7-1)^2}$

        $= \sf \sqrt{(-3)^2+6^2 } \\\\= \sqrt{9+36} \\\\= \sqrt{45} \\\\= \sqrt{9 \times 5 } \\\\= 3\sqrt{5} \ units$

$\bullet \sf \ AC = \sqrt{(-2-4)^2+(7-(-5))^2}$

       $= \sf \sqrt{(-6)^2+12^2} \\\\= \sqrt{36+144} \\\\= \sqrt{180} \\\\= \sqrt{36 \times 5 } \\\\= 6 \sqrt{5} \ units$

$\longrightarrow \sf AB+BC = 3\sqrt{5} + 3\sqrt{5} \\\\\longrightarrow AB + BC = 6 \sqrt{5} \\\\\longrightarrow \bf AB+BC = AC$

∴ The given points are collinear.