Geometry Problem Solver What does intersection mean in geometry?
The intersection of two figures is the set of points they have in common. Two lines that cross intersect at a single point. Two planes that cross intersect in a whole line. If the figures share no points at all, the intersection is empty.
What each pair of figures produces
Two distinct lines in a plane either cross at exactly one point or are parallel and never meet. They cannot cross twice: two points already determine a unique line, so two lines sharing two points are the same line.
In three dimensions there is a third option. Two lines can be skew, meaning they neither meet nor run parallel, the way a road and a flight path can pass without touching.
A line and a circle can miss entirely, touch at one point as a tangent, or cut through at two. Two circles behave the same way, meeting at zero, one or two points, unless they are the same circle.
The empty intersection
When two figures share nothing, the intersection is not undefined. It is the empty set, written ∅. Parallel lines have an empty intersection, and saying so is the correct answer rather than saying there is none.
Finding the point where two lines cross
Given two lines as equations, the intersection is the pair of values that satisfies both, so you solve them as a simultaneous system. With y = 2x + 1 and y = −x + 7, set 2x + 1 = −x + 7, which gives 3x = 6 and x = 2. Substituting back gives y = 5, so the lines cross at (2, 5).
If the algebra collapses to something impossible like 0 = 4, the lines are parallel and there is no crossing point. If it collapses to something always true like 0 = 0, they are the same line, and every point on it is shared.
What an intersection can be
| Figures | Possible intersections |
|---|---|
| Two lines in a plane | One point, or empty if parallel |
| Two lines in space | One point, empty, or skew and empty |
| Two planes | A line, or empty if parallel |
| A line and a plane | One point, a whole line, or empty |
| A line and a circle | Two points, one if tangent, or empty |
| Two circles | Two points, one if tangent, or empty |
Questions
Can two lines intersect at more than one point?
Only if they are the same line. Two points determine exactly one line, so two distinct lines sharing two points is a contradiction.
What is the intersection symbol?
∩, from set notation. A ∩ B is read as the intersection of A and B, and it means the points that belong to both.
Do parallel lines ever intersect?
Not in ordinary Euclidean geometry, which is what school courses use. In projective geometry parallel lines are defined to meet at a point at infinity, but that is a different system with different rules.
More geometry reference
Stuck on the actual problem
Photograph it and get the worked solution, step by step, with the theorem named at each one.
