Class 10

Math

All topics

Coordinate Geometry

Using the distance formula, show that the given points are collinear.

$(−2,5),(0,1)$ and $(2,−3)$.

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

Let $A(−2,5),B(0,1)$ and $C(2,−3)$

A, B and C are collinear if $AB+BC=AC$

$AB=(x_{2}−x_{1})_{2}+(y_{2}−y_{1})_{2} $

$=(0+2)_{2}+(1−5)_{2} =2_{2}+(−4)_{2} $

$=4+16 =20 $

$BC=(2−0)_{2}+(−3−1)_{2} =2_{2}+(−4)_{2} $

$=4+16 =20 $

$=sqrt4×5=25 $ units

$CA=(2+2)_{2}+(−3−5)_{2} =4_{2}+(−8)_{2} $

$=16+64 =80 =16×5 =45 $

From above, we can see that

$AB+BC=20 +25 =25 +25 =45 =AC$

Therefore, A, B and C are collinear.