Introduction to line segments collinear line
If a set of lines segments lie on the same line, they are called collinear lines. This can be verified by the following methods:
(i) If the slopes of the lines are same and if a point on one line satisfies every other line, they will be collinear.
(ii) If the sum of the distance of the end points of each line are equal to the distance of the tarting point to the ending point that are connecting all the lines.
(iii) If three end points of three lines are given, they will be collinear if their area is zero.
Now, let us few examples on line segments collinear line.
Examples on Line Segments Collinear Line
Ex 1: Let A(1,2), B( 2, 3) and C( 3, 4) be the points. Let AB, BC and CA be the line segments. Prove that they are collinear line.
Sol: The slope of the line AB is `1/1`
The slope of the line BC is `( 4 - 3)/( 3 - 2)` = `1/1`
The slope of the line CA is `( 4 - 2)/( 3 - 1)` = `2/2`
Since the slopes of all the lines AB, BC and CA are equal, they are collinear lines.
Ex 2: Let A(1,2), B( 2, 3) and C( 3, 4) be the points. Let AB, BC and CA be the line segments. Prove that they are collinear line using the area of the triangle formula.
Sol: Area of the triangle formula is `1/2` [ x1( y2 - y3) + x2( y3 - y1) + x3 ( y1 - y2)].
= `1/2` [ 1( 3 - 4) + 2 ( 4 - 2) + 3 ( 2 - 3)]
= `1/2` [ ( -1) + 2(2) + 3(-1)]
= `1/2` ( - 4 + 4 ) = 0.
Since the area of the triangle is zero, the points A , B and C will lie on a straight line.
Ex 3: Let A(1,2), B( 2, 3) and C( 3, 4) be the points. Let AB, BC and CA be the line segments. Prove that they are collinear line using distance formula.
Sol:
The distance between AB is:
The distance between BC is:
The distance between AC is:
Here the distance between A and C is equal to the sum of the distances between A,B and B,C.
Therefore the three points are collinear.
I hope the above discussion on line segments collinear line would help you understand the concept very well.
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Practice Problems on Line Segments Collinear Line
1. Check whether the points A(2,5), B(3. 7) and C( 2, 3) are collinear. [ Answer: They are not collinear.]
2. Check whether the points A(1,3), B(3,5) and C( 5,7) are collinear. [ Answer: They are collinear.]
If a set of lines segments lie on the same line, they are called collinear lines. This can be verified by the following methods:
(i) If the slopes of the lines are same and if a point on one line satisfies every other line, they will be collinear.
(ii) If the sum of the distance of the end points of each line are equal to the distance of the tarting point to the ending point that are connecting all the lines.
(iii) If three end points of three lines are given, they will be collinear if their area is zero.
Now, let us few examples on line segments collinear line.
Examples on Line Segments Collinear Line
Ex 1: Let A(1,2), B( 2, 3) and C( 3, 4) be the points. Let AB, BC and CA be the line segments. Prove that they are collinear line.
Sol: The slope of the line AB is `1/1`
The slope of the line BC is `( 4 - 3)/( 3 - 2)` = `1/1`
The slope of the line CA is `( 4 - 2)/( 3 - 1)` = `2/2`
Since the slopes of all the lines AB, BC and CA are equal, they are collinear lines.
Ex 2: Let A(1,2), B( 2, 3) and C( 3, 4) be the points. Let AB, BC and CA be the line segments. Prove that they are collinear line using the area of the triangle formula.
Sol: Area of the triangle formula is `1/2` [ x1( y2 - y3) + x2( y3 - y1) + x3 ( y1 - y2)].
= `1/2` [ 1( 3 - 4) + 2 ( 4 - 2) + 3 ( 2 - 3)]
= `1/2` [ ( -1) + 2(2) + 3(-1)]
= `1/2` ( - 4 + 4 ) = 0.
Since the area of the triangle is zero, the points A , B and C will lie on a straight line.
Ex 3: Let A(1,2), B( 2, 3) and C( 3, 4) be the points. Let AB, BC and CA be the line segments. Prove that they are collinear line using distance formula.
Sol:
The distance between AB is:
Here the distance between A and C is equal to the sum of the distances between A,B and B,C.
Therefore the three points are collinear.
I hope the above discussion on line segments collinear line would help you understand the concept very well.
Stuck on any of these topics how to factor the polynomial, math homework help online for free try out some best online tutoring math website.
Practice Problems on Line Segments Collinear Line
1. Check whether the points A(2,5), B(3. 7) and C( 2, 3) are collinear. [ Answer: They are not collinear.]
2. Check whether the points A(1,3), B(3,5) and C( 5,7) are collinear. [ Answer: They are collinear.]
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