Determine whether the points A(-1, -1), B (0, 1) and C (1, 3) are collinear
Answers
Step-by-step explanation:
Given:-
A(-1, -1), B (0, 1) and C (1, 3)
To find:-
Determine whether the points A(-1, -1), B (0, 1) and C (1, 3) are collinear points or not?
Solution:-
Method :-
Given points are A(-1, -1), B (0, 1) and
C (1, 3)
To show the points A ,B and C are collinear. then we have to show that AB+BC = AC.
Distance between A (x1, y1) and (x2, y2) is
√[(x2-x1)^2 + (y2-y1)^2] units
Distance between A and B:-
(x1, y1)=(-1,-1)=>x1 = -1 and y1 = -1
(x2, y2)=(0,1)=>x2 = 0 and y2 = 1
Distance between A (x1, y1) and (x2, y2) is
√[(x2-x1)^2 + (y2-y1)^2] units
=>√[{0-(-1)}^2 + {1-(-1)}^2]
=>√[(0+1)^2 +(1+1)^2]
=>√[1^2+2^2]
=>√(1+4)
=>√5 units
AB=√5 units --------------------(1)
Distance between B and C :-
(x1, y1)=(0,1)=>x1 = 0 and y1 = 1
(x2, y2)=(1,3)=>x2 = 1 and y2 = 3
Distance between A (x1, y1) and (x2, y2) is
√[(x2-x1)^2 + (y2-y1)^2] units
=>√[(1-0)^2+(3-1)^2]
=>√[1^2+2^2]
=>√(1+4)
=>√5 units
BC = √5 units ------------------(2)
Distance between A and C:-
(x1, y1)=(-1,-1)=>x1 = -1 and y1 = -1
(x2, y2)=(1,3)=>x2 = 1 and y2 = 3
Distance between A (x1, y1) and (x2, y2) is
√[(x2-x1)^2 + (y2-y1)^2] units
=>√[{1-(-1)}^2 +{3-(-1)}^2]
=>√[(1+1)^2+(3+1)^2]
=>√[2^2+4^2]
=>√(4+16)
=>√20
=>√(2×2×5)
=>2√5 units
AC = 2√5 units ------------------(3)
From (1) ,(2) & (3)
AB+BC = √5 + √5 = 2√5
AB+BC = CA
The points A(-1, -1), B (0, 1) and C (1, 3) are collinear points.
Method -2:-
To show that given points are collinear then we have to show that the area of a triangle formed by the given points is zero.
Given points are A(-1, -1), B (0, 1) and
C (1, 3)
(x1, y1)=(-1,-1)=>x1=-1 and y1 = -1
(x2, y2)=(0,1)=>x2=0 and y2=1
(x3,y3)=(1,3)=>x3 = 1 and y3 =3
We know that
The area of the triangle formed by the points (x1, y1);(x2, y2) and (x3, y3) is
(1/2) | x1(y2-y3)+x2(y3-y1)+x3(y1-y2) | sq.units
=>(1/2) | (-1)(1-3)+(0)(3-(-1))+(1)(-1-1) |
=>(1/2) | (-1)(-2)+0(3+1)+(-2) |
=>(1/2) | 2+0-2 |
=>(1/2) | 0|
=>(1/2)×0
=>0/2
=>0
Area of the triangle is 0
Given points A,B and C are collinear points.
Answer:-
The points A(-1, -1), B (0, 1) and C (1, 3) are collinear points
Used formulae:-
- Distance between A (x1, y1) and (x2, y2) is
- √[(x2-x1)^2 + (y2-y1)^2] units
- The area of the triangle formed by the points (x1, y1);(x2, y2) and (x3, y3) is
- (1/2) | x1(y2-y3)+x2(y3-y1)+x3(y1-y2) | sq.units
- The points A,B,C are collinear points if AB+BC = AC.
- The points on the same line are called Collinear points.
- If the Area of a triangle formed by the three points is zero then the points are Collinear points.