Question

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

Answer 1

Points are collinear if the sum of any two distances is equal to the distance of the third.

Let $A(1,-1),B(5,2),C(9,5).$

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

$=\sqrt{4^2+3^2}$

$=\sqrt{16+9}$

$=\sqrt{25}$

$=5\text{units}$

$BC=\sqrt{(9-5{)}^2+(5-2{)}^2}$

$=\sqrt{4^2+3^2}$

$=\sqrt{16+9}$

$=\sqrt{25}$

$=5\text{units}$

$AC=\sqrt{(9-1{)}^2+(5+1{)}^2}$

$=\sqrt{8^2+6^2}$

$=\sqrt{64+36}$

$=\sqrt{100}$

$=10\text{units}$

From above, we can conclude that

$AB+BC=AC$

Therefore, points $A,B$ and $C$ are collinear.