Question

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

Answer 1

Answer - Yes, the Points are Collinear. 

Explanation -
Let the Points A(1,-3), B(2,-5), C(-4,7) be A(x₁, y₁), B(x₂,y₂), C(x₃,y₃).

Let us first find the Slope of AB,

∵ m = $\frac{y_2 - y_1}{x_2 - x_1}$
∴ m = (-5 + 3)/(2 - 1)
    = -2/1
   =-2

Now For th Slope of BC, 
m = $\frac{y_3 - y_2}{x_3 - x_2}$
   = (7 + 5)/(-4 - 2)
  = 12/-6
  = -2

Since, the Slope of both the lines AB, and BC are same therefore, Points are Collinear. 


Hope it helps.

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 .

Area∆ABC

= 1/2|x1(y2-y3)+x2(y3-y1)+x3(y2-y1)|

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

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

= 1/2| -12 + 2×10 + (-4)(2) |

= 1/2 | -12 + 20 - 8 |

= 1/2 | 20 - 20 |

= 1/2 × 0

= 0

Therefore ,

area ∆ABC = 0 ,

A, B and C are collinear .

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