plz give me the answer
Answers
the points are collinear.......
NOTE:- if u don't believe u can check by marking points on a graph paper
DON'T FORGET TO MARK IT AS THE BRAINLIEST............
Given :
Points : A(1 , -3) , B(2 , -5) , C(-4 , 7) .
To determine :
Whether the given points are collinear .
Approach 1 :
Using distance formula :
• Note : The distance between the points A(x1 , y1) and B(x2 , y2) is given by ;
AB = √ [ (x2 - x1)² + (y2 - y1)² ]
Thus ,
The distance between points A(1 , -3) and B(2 , -5) will be ;
=> AB = √[ (2 - 1)² + (-5 + 3)² ]
=> AB = √[ 1² + (-2)² ]
=> AB = √[ 1 + 4 ]
=> AB = √5
Also ,
The distance between points B(2 , -5) and C(-4 , 7) will be ;
=> BC = √[ (-4 - 2)² + (7 + 5)² ]
=> BC = √[ (-6)² + (12)² ]
=> BC = √[ 36 + 144 ]
=> BC = √180
=> BC = 6√5
Also ,
The distance between points A(1 , -3) and C(-4 , 7) will be ;
=> AC = √[ (-4 - 1)² + (7 + 3)² ]
=> AC = √[ (-5)² + (10)² ]
=> AC = √[ 25 + 100 ]
=> AC = √125
=> AC = 5√5
Now ,
=> AB + AC = √5 + 5√5
=> AB + AC = 6√5
=> AB + AC = BC
=> The points A , B and C are collinear .
[ Note : From the equation AB + AC = BC , we can conclude that point A lies between points B and C . ]
Approach 2 :
Using slope of straight line :
• Note : The slope of the line joining the points A(x1 , y1) and B(x2 , y2) is given by ;
Slope of AB = (x2 - x1) / (y2 - y1)
Thus ,
The slope of line segment joining the points A(1 , -3) and B(2 , -5) will be ;
=> Slope of AB = (-5 + 3)/(2 - 1)
=> Slope of AB = -2/1
=> Slope of AB = -2
Also ,
The slope of line segment joining the points A(1 , -3) and C(-4 , 7) will be ;
=> Slope of AC = (7 + 3)/(-4 - 1)
=> Slope of AC = 10/-5
=> Slope of AC = -2
Clearly ,
Slope of AB = Slope of AC .
Thus ,
The points A , B and C are collinear .
Approach 3 :
Using area of triangle :
• Note : The area of the triangle formed by the points A(x1 , y1) , B(x2 , y2) and C(x3 , y3) is given by ;
∆ = ½•| x1(y2-y3) + x2(y3-y1) + x3(y1-y2) |
Thus ,
The area of triangle formed by the points A(1 , -3) , B(2 , -5) and C(-4 , 7) will be ;
=> ∆ = ½•| 1(-5-7) + 2(7+3) + (-4)(-3+5) |
=> ∆ = ½•| 1•(-12) + 2•10 + (-4)•2 |
=> ∆ = ½•| -12 + 20 - 8
=> ∆ = ½•| 20 - 20 |
=> ∆ = ½•| 0 |
=> ∆ = ½•0
=> ∆ = 0
Since the area of the triangle formed by the points A , B and C is zero , thus the points A , B and C are collinear .