parallel lines are two lines that never meet. find an example that contradicts this definition. How would you change the definition to make it more accurate?
Answers
EXPLANATION
By definition two lines are said to be parallel when the distance between two lines is same when measured from any point on one line.
However the criteria for intersection is that the distance between two lines should change and eventually should change to zero at intersection point.
ADVANCED EXPLAINATION
Two lines are said to be parallel when their slopes are equal.
Every line can be written in a generalized form as
y= mx+c
Where x, y are two coordinates in axis system.
C is intercept of line on y axis ( coordinate where the line intersects X axis)
M is slope of line, it indirectly tells us inclination (Theta 'θ' : it is the angle that the line makes from positive x axis) .
value of m can be determined as
m=tan θ
Thus if lines are parallel both line equations will have same slope that is same angle (inclination) with x axis.
For example let's assume one equation of line y=6x+3. You can draw line on graph using equation of line. You can get any point on line by substituting any assumed value of x in above equation and calculating corresponding value of y. If you put x=1 you will get value of y=6×1+3=9.
Now assume any other line that is parallel to above line. You can change the value of c from above equation and you can get as many parallel line equations as you need.
Let's assume y = 6x + 5
If you want to find if these lines intersect, the intersection point will satisfy both equations of line.
If you try to solve both equations of line for value of x and y using method of simultaneous equations, you will notice that there is no solution to these equations.
This means that these two lines don't have any point in common.
Thus parallel lines will never intersect.
Why do parallel lines never intersect?
Simple explanation
By definition two lines are said to be parallel when the distance between two lines is same when measured from any point on one line.
However the criteria for intersection is that the distance between two lines should change and eventually should change to zero at intersection point.
Advanced and unnecessary explanation ;P
Two lines are said to be parallel when their slopes are equal.
Every line can be written in a generalized form as
y= mx+c
Where x, y are two coordinates in axis system.
C is intercept of line on y axis ( coordinate where the line intersects X axis)
M is slope of line, it indirectly tells us inclination (Theta: it is the angle that the line makes from positive x axis) .value of m can be determined as
m=tan (THETA)
Thus if lines are parallel both line equations will have same slope that is same angle (inclination) with x axis.
For example let's assume one equation of line y= 6x+3. You can draw line on graph using equation of line. You can get any point on line by substituting any assumed value of x in above equation and calculating corresponding value of y. If you put x=1 you will get value of y=6×1+3=9.
Now assume any other line that is parallel to above line. You can change the value of c from above equation and you can get as many parallel line equations as you need.
Let's assume y = 6x + 5
If you want to find if these lines intersect, the intersection point will satisfy both equations of line.
If you try to solve both equations of line for value of x and y using method of simultaneous equations, you will notice that there is no solution to these equations.
This means that these two lines don't have any point in common.
Thus parallel lines will never intersect.