Question

determine weather the following are colinear A(2,-5) ,B(2,-5) C(-4,7)​

Answer 1

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$\bold{Explanation:}$

A(2,-5),B(2,-5) ,C(-4,7)

$\sf x_{1}=2,x_{2}=2 \&\: x_{3}=-4$

$\sf y_{1}=-5,y_{2}=-5\&\: y_{3}=7$

$\sf AB= \sqrt{ {(x_{2} - x_{1})}^{2} + {(y_{2 }- y_{1})}^{2} }$

$\sf = > AB = \sqrt{ {(2 - 2)}^{2} + {( - 5 + 5)}^{2} } = 0$

$\sf BC= \sqrt{ {(x_{3} - x_{2})}^{2} + {(y_{3 }- y_{2})}^{2} }$

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

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

$\sf CA= \sqrt{ {(x_{3} - x_{1})}^{2} + {(y_{3 }- y_{1})}^{2} }$

$\sf =>CA = \sqrt{ {( - 4 - 2)}^{2} + {(7 + 5)}^{2} } = \sqrt{180} = 6 \sqrt{5}$

Since ,AB+BC=CA

$\sf\bold{0+6 \sqrt{5}=6\sqrt{5}}$

Here ,we see that sum of AB and BC is equal to the CA.

Hence, we can say that above points are collinear.