# Rays, points and line segments

• Lee33
In summary, the question asks if the point D is not in the ray ##\vec{AB}## then prove that {A}. But there's no C in the question, so this is not possible.
Lee33

## Homework Statement

Let A, B and D be points. If ##D\notin \vec{AB}## in a metric geometry, prove that ##\vec{AD}\cap\vec{AB} = \{A\}.##2. Relevant definitions

Ray: If A and B are distinct points then the ray from A toward B is the set ##\vec{AB} = \overline{AB}
\cup \{ C \in P \ | \ A-B-C\},## where ##A-B-C## means B is between A and C.

Line segment: If A and B are distinct points in a metric geometry then the line segment from A to B is the set ##\overline{AB} = \{C\in P\ | \ A-C-B \ \text{or} \ C=A \ \text{or} \ C = B\}.##

## The Attempt at a Solution

I am not sure how to go about proving this.

If I let ##Z## be a point where ##Z\in\vec{AD}## then ##Z\notin\vec{AB}## since ##Z## was arbitrary then there are no points in ##\vec{AD}## that lie in ##\vec{AB}##. So ##A## is the only point in ##\vec{AD}\cap\vec{AB}##.

Hi Lee33!
Lee33 said:
If I let ##Z## be a point where ##Z\in\vec{AD}## then ##Z\notin\vec{AB}## since ##Z## was arbitrary then there are no points in ##\vec{AD}## that lie in ##\vec{AB}##. So ##A## is the only point in ##\vec{AD}\cap\vec{AB}##.

Haven't you just assumed exactly what you have to prove?

And don't say "Z is arbitrary", it's almost meaningless … say "if Z ≠ A then A – Z – D"

(However, I don't understand the question … why can't D be along the extended line AB, the other side of B ??)

Well the question is asking if the point D is not in the ray ##\vec{AB}## then I must prove that ##\vec{AB}\cap\vec{AD}## equals the set {A}.

Last edited:
but there's no C in the question
Lee33 said:
Let A, B and D be points. If ##D\notin \vec{AB}## in a metric geometry, prove that ##\vec{AD}\cap\vec{AB} = \{A\}.##

Opps. I meant A instead of C. I have edited the post. Sorry about that.

## 1. What is the difference between a ray and a line segment?

A ray is a part of a line that starts at one endpoint and extends infinitely in one direction. A line segment is a part of a line that has two endpoints and a fixed length.

## 2. How do you name a ray?

A ray is typically named by its endpoint and another point on the ray. For example, if a ray starts at point A and extends through point B, it can be named as "ray AB" or "ray BA".

## 3. Can a point be part of a line segment?

Yes, a point can be one of the endpoints of a line segment.

## 4. How many rays can be drawn from a single endpoint?

Infinitely many rays can be drawn from a single endpoint, in any direction.

## 5. Are rays, points, and line segments considered as two-dimensional or three-dimensional objects?

Rays, points, and line segments are considered as two-dimensional objects as they only have length and no width or depth.

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