Two lines intersect at three points?

  • Context: High School 
  • Thread starter Thread starter Mr. X
  • Start date Start date
  • Tags Tags
    Lines Points
Click For Summary
SUMMARY

The discussion centers on the intersection of the lines y=x and y=2x, concluding that they only intersect at the point (0,0) in Euclidean geometry. The initial claim of three intersection points arises from substituting infinity and negative infinity into the equations, which are not considered real numbers. The conversation highlights the distinction between Euclidean geometry, where lines intersect at a single point, and other geometries like spherical or hyperbolic geometry, where multiple intersections can occur.

PREREQUISITES
  • Understanding of linear equations and their graphical representation.
  • Familiarity with the concept of intersection points in geometry.
  • Knowledge of Euclidean geometry principles.
  • Basic comprehension of alternative geometries such as spherical and hyperbolic geometry.
NEXT STEPS
  • Study the properties of linear equations in Euclidean geometry.
  • Explore the concept of infinity in mathematical contexts.
  • Learn about spherical geometry and its unique intersection properties.
  • Investigate hyperbolic geometry and how it differs from Euclidean geometry.
USEFUL FOR

Students of mathematics, educators teaching geometry, and anyone interested in the properties of lines and intersections in various geometrical contexts.

Mr. X
Messages
2
Reaction score
0
Two lines intersect at three points?

I have a question that's been bugging my mind. :confused: Imagine the lines y=x and y=2x. Let's find their intersection point.
y=x and
y=2x so that
x=2x If we give x the value of 0 the equation will be correct. (1st intersection)
But if we give x the value of infinity the equation will be correct once again.
(inf=2*inf) (2nd intersection)
If we put negative infinity for x the equation will be correct. (3rd equation)
We have found three intersection points. We know that lines either do not intersect or intersect at only one point or intersect at infinite points. They certainly cannot intersect at three points. Is it because infinity is not a real number, which means that we don't care about the points of intersection whose coordinates are equal to either infinity or negative infinity? :wink:
 
Mathematics news on Phys.org
I assume x and y are real numbers? In which case you can not substitute in infinity because it is not a real number.
 
Mr. X said:
I have a question that's been bugging my mind. :confused: Imagine the lines y=x and y=2x. Let's find their intersection point.
y=x and
y=2x so that
x=2x If we give x the value of 0 the equation will be correct. (1st intersection)
But if we give x the value of infinity the equation will be correct once again.
(inf=2*inf) (2nd intersection)
If we put negative infinity for x the equation will be correct. (3rd equation)
We have found three intersection points. We know that lines either do not intersect or intersect at only one point or intersect at infinite points. They certainly cannot intersect at three points. Is it because infinity is not a real number, which means that we don't care about the points of intersection whose coordinates are equal to either infinity or negative infinity? :wink:

In Euclidean (planar) geometry, there are no points at infinity - so the two lines only meet at x=y=0.

There are other geometries where lines can meet in more than one place. For example, spherical or hyperbolic geometry.
 

Similar threads

High School Circle tangent to two lines and another circle
  • · Replies 2 ·
Replies
2
Views
6K
MHB Finding Points of Intersection for Trigonometric Functions
  • · Replies 8 ·
Replies
8
Views
3K
High School Invariant under rotation: Banal, obvious, or noteworthy?
  • · Replies 2 ·
Replies
2
Views
2K
MHB Quadratic Equation Help: Find 3 Points to Determine Answer
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Family of Circles at Two Points
  • · Replies 2 ·
Replies
2
Views
2K
High School How would I write this trig solution?
  • · Replies 19 ·
Replies
19
Views
3K
MHB Planes 1 & 2 Intersect: Find Scalar Equations
  • · Replies 1 ·
Replies
1
Views
2K
MHB Line of intersection between two parallel planes?
  • · Replies 7 ·
Replies
7
Views
5K
MHB Where Do the Diagonals of a Parallelogram Intersect?
  • · Replies 8 ·
Replies
8
Views
4K
MHB How Are Tangents to a Circle Found and Their Intersection Point Determined?
  • · Replies 5 ·
Replies
5
Views
2K