Question

Determine whether the points are collinear.
A(1,-3)
B,(2,-5)
C(-4,7) ​

Answer 1

Answer:

the value is not colinear become the area of triangle is 6

Step-by-step explanation:

$1 \div 2 \times ( - 5 + 14 + 12) - (6 + 20 + 7) \\ by \: using \: the \: area \: of \: triangle \: formula \\ = 6$

hence not collinear

Answer 2

$Let\:,\\A(1, - 3) = ( \: x_{1} \: \: y_{1}) \\ B (2,-5) = ( x_{2}\: \: y_{2}) \\C (-4,7) = ( x_{3} \: \:y_{3})$

$Slope \: of \: line \: AB = \dfrac{ y_{2} - y_{1} }{ x_{2} - { x_{1} }} \\ \\= \dfrac{ - 5 - ( - 3)}{2 - 1} \\ \\ = \dfrac{ - 5 + 3}{1} \\ \\ = - 2$

Slope of line AB = - 2 ...... equation 1

$Slope \: of \: line \: BC = \dfrac{ y_{3} - y_{2} }{ x_{3} - { x_{2} }} \\ \\ = \dfrac{ 7 - ( - 5)}{-4 - 2} \\ \\ = \dfrac{ 7 + 5}{-6} \\ \\ = \dfrac{ 12 }{-6} \\ \\ = - 2$

Slope of line BC = - 2 ..... equation 2

Slope of line AB = Slope of line BC ( From 1 and 2 )

Therefore , line AB and line BC are collinear to each other.