Show that the point $A( 4,\ 2),\ B( 7,\ 5),\ C( 9,\ 7)$ are collinear.

Given: Point $A( 4,\ 2),\ B( 7,\ 5),\ C( 9,\ 7)$.

To do: To show that the given points are collinear.

Solution:

Given points are: $A(4,\ 2),\ B( 7,\ 5),\ C( 9,\ 7)$

$AB=( 4, 2)\ ( 7,\ 5)$

$x_1=4,\ y_1=2,\ x_2=7,\ y_2=5$

Slope for $AB=\frac{y_2-y_1}{x_2-x_1}$

$=\frac{5-2}{7-4}$

$=\frac{3}{3}$

​

$=1$

For points $B( 7,\ 5)$ and  $C( 9,\ 7)$

$x_1=7,\ y_1=5,\ x_2=9,\ y_2=7$

Slope for $BC=\frac{7-5}{9-1}$

​

$=\frac{2}{2}$

​

$=1$

For points $C( 9,\ 7)$ and $A( 4,\ 2)$

$x_1=9,\ y_1=7,\ x_2=4,\ y_2=2$

Slope for $CA=\frac{2-7}{4-9}$

​

$=\frac{-5}{-5}$

​

$=1$

Hence, slope for $AB=$ slope for $BC=$ slope for $CA$.

Hence, it has been proved that the given points $A( 4,\ 2),\ B( 7,\ 5),\ C( 9,\ 7)$ are collinear.

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Updated on: 10-Oct-2022

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