Question

Determine whether the points are collinear.
(1) A(3,-3), B(2,-5), C(-4,8)​

Answer 1

$\huge{\bf{\green{\mathfrak{Question:-}}}}$

• Determine whether the points are collinear.
• (1) A(3,-3), B(2,-5), C(-4,8)

$\huge {\bf{\orange{\mathfrak{Answer:-}}}}$

• Given points are A (3 , - 3) ; B (2 , - 5) and C ( - 4 , 8)

• We know that points are collinear if the area of triangle = 0.

• $\textsf{1/2 \: |\: x1 \: ( \: y2 \: - y3 \: ) \: + x2 \: ( \: y3 \: - \: y1 \: ) \: + \: x3 \: ( \: y1 \: - \: y2 \: ) \: | \: = \: 0}$

• $\textsf{1/2\: | \: 3 \: ( \: -5 \: - \: 8 \: ) \: + \: 2 \: ( \: 8 \: - \: (-3) \: ) \: + \: -4 \: ( \: -3 \: - \: (-5) \: ) \: |}$

• $\textsf{1/2 \: | \: 3 \: ( \: -13 \: ) \: + \: 2 \: ( \: 11 \: ) \: + \: -4 \: ( \: 2 \: ) \: |}$

• $\textsf{1/2 \: | \: -39 \: + \: 22 \: + \: -8 \: |}$

• $\textsf{1/2 \: | \: -25 \: |}$

• $\textsf{1/2 \: * \: 25}$

• $25/2 \: ≠ \: 0$

$\huge{\bf{\red{\mathfrak{Conclusion:-}}}}$

• Therefore, A(3,-3), B(2,-5), C(-4,8) are not collinear.

Answer 2

Let A(1,-3)=(x1.y1), B(2,-5)-(x2.y2),

and C(-4,7) = (x3.y3) are three veriticies

of a Triangle ABC.

AreaAABC

= 1/2lx1(y2-y3)+x2(y3-y1)+x3(y2-y1)l

=1/211(-5-7]+2[7-(-3)]+(-4)[-3+5]|

= 1/21 (-12)+2(7+3)+(-4)(-3+5)|

= 1/21 -12 + 2x1O + (-4)(2)I

= 1/21 -12 + 20 - 81

= 1/2 | 20 - 20I1

= 1/2 x 0

= 0

Therefore,

area AABC = 0,

A, B and C are collinear.