10. Vishnu said that the points (1, 2), (2, 2) and (3, 2) are collinear. Priya said that the points are not collinear. The correct statement given by [ ]
Answers
Answer:
the correct statement is given by VISHNU
Step-by-step explanation:
for three points to be collinear, the area of the triangle formed by these three points has to be 0
area of triangle = 1/2 ( x₁ [ y₂ - y₃ ] + x₂ [ y₃ - y₁ ] + x₃ [y₁ - y₂ ] )
x₁ = 1 , x₂ = 2 , x₃ = 3
y₁ = 2 , y₂ = 2 , y₃ = 2
area of triangle = 1/2 ( 1 [ 2 - 2 ] + 2 [ 2 - 2 ] + 3 [2 - 2 ] )
= 1/2 ( 1 [ 0 ] + 2 [ 0 ] + 3 [ 0 ] )
= 1/2 ( 0 + 0 + 0 )
= 1/2 x 0
= 0
so the area of the triangle is zero which means the three points are collinear
therefore the correct statement is given by VISHNU
Explanation:
We can check whether these points are collinear or not by using various methods. I am mentioning two methods here.
METHOD 1 : By DIstance Formula
In general, three points A, B and C are collinear if the sum of the lengths of any two line segments among AB, BC and CA is equal to the length of the remaining line segment, that is, either AB + BC = AC or AC +CB = AB or BA + AC = BC.
The basic formula for this method is sqrt { (x2 - x1)^{2} + (y2 - y1)^{2} }
∴ Taking (1,2) as (x1, y1) and (2,2) as (x2,y2) ,
DIstance between the points = sqrt { (2 - 1) ^2 + (2 - 2) ^2 }
= sqrt { 1 }
= 1
Now taking (2,2) as (x1, y1) and (3,2) as (x2, y2) ,
Distance between the points = sqrt { (3 - 2) ^2 + (2 - 2) ^2 }
= sqrt { 1 }
= 1
Now taking (3,2) as (x1, y1) and (1,2) as (x2, y2) ,
Distance between the points = sqrt { (1 - 3) ^2 + (2 - 2) ^2 }
= sqrt { 4 }
= 2
∴ According to the principle, the sum of distance between (1,2) , (2,2) and
(2,2) , (3,2) is equal to the distance between (3,2) , (1,2) .
∴ The three points (1, 2), (2, 2) and (3, 2) are collinear using distance
formula.
∴ The correct statement was given by Vishnu.
METHOD 2 : By Slope
This method is mainly used in equation of lines. Three points A, B and C are said to be collinear if the slopes of all the lines are equal, that is, if the slope of AB, the slope of BC and the slope of CA are equal, the points are collinear.
The general formula of finding the slope is :
Slope = (y2 - y1) / (x2 - x1)
∴ Taking (1,2) as (x1, y1) and (2,2) as (x2,y2) ,
Slope = (2 - 2) / (2 - 1)
= 0 / 1
= 0
Now taking (2,2) as (x1, y1) and (3,2) as (x2, y2) ,
Slope = (2 - 2) / (3 - 2)
= 0 / 1
= 0
Now taking (3,2) as (x1, y1) and (1,2) as (x2, y2) ,
Slope = (2 - 2) / (1 - 3)
= 0 / -2
= 0
∴ According to the principle, the slope between (1,2) and (2,2) , the slope between (2,2) and (3,2) and the slope between (3,2) and (1,2) are equal, that is, 0.
∴ The points (1, 2), (2, 2) and (3, 2) are collinear.
∴ The correct statement was given by Vishnu.
I hope this answer helps you.