Rays in mathematics are part of Geometry that has one side fixed and the other can be infinitely extended. In geometry, a ray is a line that starts at a single point, known as the endpoint and stretches infinitely in one direction. A real-world example of a ray is a beam of sunlight; the sun serves as the endpoint, and the light travels outward without end.

The representation of Ray is different from the other representations of line and line segments and holds a special significance for its unique properties.

In this article, we will discuss the real meaning of ‘Rays’ in geometry followed by ray properties, representation, real-life examples, and differences between rays and other representations like line and line segments. At the end of this blog, we will also look at some solved examples and practice problems on rays in geometry.

What is Ray in Geometry?

Ray in Geometry is a special type of representation that can be infinitely extended from one side while the other side is static and represented by a single endpoint.

Ray can thus be considered as a one-side infinitely extended geometrical representation. In this blog, we have described ‘Ray in geometry’ in a detailed manner.

Ray in Mathematics Definition

Ray in Mathematics is any line that has a fixed endpoint on one side and on the other side, it can be extended infinitely.

The fixed endpoint is depicted by a point whereas the infinitely extandible side has an arrow on it.

Ray Notation

As ray only extends infinitely in one direction. To name a ray, say the name of the endpoint and any other point on the ray. The symbol for ray is →.

For example, if a ray has endpoint A, and B is any other point on the half-line, then the ray will be denoted by [Tex]\overrightarrow{AB}[/Tex]

Ray in Geometry Examples in Real Life

The real world example of ray would be considering the sunlight originating from their source sun as the fixed endpoint and reaching earth with sun-rays that can be infinitely extended in one direction. The sun is considered as static fixed point and rays constitute the infinitely extendible side of ray.

Properties of Ray

Ray is fundamental to the geometry and vectors. Hence, we must learn its unique properties. The properties of Ray are discussed below:

• A Ray only has a single endpoint on one side of it. The single endpoint is depicted by a point on one of the side of it.
• Other side of ray can be infinitely extended and has no fixed end point. The other side of ray which can be infinitely extended is represented by arrow symbol.
• Since, Ray can be extended indefinitely on the one side we can’t measure its exact length.
• Two rays are only equal only they have same endpoint and extended in the same direction.

Representation of Ray in Geometry

We know that a Ray has one end point and it can be extended only in one direction. Let we have a Ray AB in which A is the end Point and it can be extended in the direction of B. The image of ray AB is attached below:

Now let’s take another Ray BA in which B is the end point and it can be extended in the direction of A. The image of ray BA is attached below:

Now one can think of that Ray AB is Same as Ray BA but it is wrong. Ray AB is Not equal to Ray BA as in among them the endpoints are different and are extended in opposite directions. Hence, two rays are only similar if they have same endpoint and extended in the same direction.

Above image shows a ray where point A is the static point of ray which is fixed while point B is the one which can be infinitely extended as per need.

Ray and Angles

Angles are formed from rays in geometry. An angle is formed when two rays combine with their fixed endpoints overlapping and the other infinitely extended side of ray then represent the arm of an angle.

Angle is the bent produced or the region between the two rays when they meet. In mathematics terms, the two rays that meet to form angle are called Arms of the Angle. One should note that the arms of angle are rays hence they can be extended individually in their defined direction and the extension of arms of angle doesn’t affect the value of the angle.

In the above figure represents ‘O’ as the fixed endpoint of two rays and OB and OA as the two individual rays making together an angle.

Learn More, Types of Angles

Line, Line Segment and Rays

Line, Line Segment and Rays are three fundamentals of geometry which forms the basis of almost every geometrical shape and figures. Let’s learn about their basic definitions and comparison between them.

• Line: A Line in mathematical geometry can be extended in both directions.
• Line Segment: A Line segment in mathematical geometry cannot be extended in any directions. It is represented by two fixed end points.
• Ray: A Line in mathematical geometry can be extended only in one direction.

Differences between Line, Line segment and Ray

The differences between Line, Line Segment and Ray is tabulated below:

Line	Line Segment	Ray
Line has as no endpoint.	Line segment has two endpoints.	Ray has one endpoint.
Both sides can be infinitely extended.	No side of line segment can be extended.	One side of a ray can be infinitely extended.
A line is represented by arrows on both sides.	A line segment is represented by two fixed points on both sides.	A ray is represented by an arrow on one side and fixed point on one side.

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Solved Examples on Rays

Below are some examples to help you understand about rays in geometry.

Example 1: Draw a ray of length 10 cm and label the fixed endpoint as P and point that can extended indefinItely as Q.

Example 2: Identify the fixed endpoint and point which can be infinitely extended for the below ray image.

Solution:

In the above image,

Point R is the fixed endpoint which is static and cannot be extended as it is represent by point.

Point S is the point which can be infinitely extended as it is represented by arrow.

Example 3: Identify the point which can be infinitely extended for the below ray image.

Solution:

In the above image, point B can be infinitely extended for the ray.

Example 4: Draw a ray of 5 cm and represent the fixed endpoint as S.

Solution:

Here, S is the fixed point, T is the point which can be extended infinitely and the ray length is 5 cm.

Example 5: Draw a ray of 7 cm and represent the fixed endpoint as Y and the infinitely extendible point as Z.

Solution:

Here, Y is the fixed point, Z is the point which can be extended infinitely and the ray length is 7 cm.

Practice Question on Ray in Geometry

Q1: Draw a ray of 5 cm in length. Name the ray points of both direction.

Q2: Draw a ray of 7 cm and represent the fixed endpoint as P.

Q3: Draw a ray of 9 cm and name the fixed endpoint as R and the extendible point as S.

Q4: Draw a ray of 15 cm and represent the fixed endpoint as A and and the extendible point as B.

Q5: Draw a line, line segment and ray each of 6 cm and label both sides of all three.

Q6: Draw a line, line segment and ray each of 8 cm and label sides of line with A-B, line segment with C-D and ray with E-F points.

Summary – Ray in Geometry

In geometry, a ray is a line with a fixed starting point on one end and the ability to extend infinitely in the other direction. This is different from a line, which extends in both directions, and a line segment, which does not extend at all and has two fixed endpoints. Rays are important in geometry and are used to form angles when two rays originate from the same endpoint and extend outward. The concept of rays is also applicable in real life, such as sunlight traveling from the sun to the Earth, with the sun acting as the fixed endpoint. Understanding rays helps in studying various properties and their applications in both theoretical and practical scenarios in geometry.

FAQs on Rays in Geometry

What is Ray in maths?

Ray in Mathematics are lines with one side having fixed endpoint and the other side as infinitely extendible side represented by an arrow.

How to Represent Rays?

Rays are represented by a fixed endpoint on one side and an arrow on the other side.

What is Difference Between Line and Ray?

A line can be infinitely extended in both directions whereas a ray can infinitely extended only in one direction.

What is Difference between Line segment and Ray?

A line segment has two fixed endpoints and cannot be extended while a ray has only one fixed endpoint and can be infinitely extended from one side.

What are the Two sides of an Angle made up of: line, line segment or ray?

The two sides of an angle are made up of two individual rays.

Can a Ray Be Defined by Two Points?

No, a ray cannot be defined by two points. It’s defined by one endpoint and extends infinitely in one direction. Two points determine a line segment.

What is the Name of the Endpoint of a Ray?

The endpoint of a ray is called the “origin” or “initial point.”

Can Rays Intersect or Cross Each Other?

Rays can intersect or cross if their extensions meet at a point. However, they do not cross if they share only a common endpoint and extend in different directions.

What Are Some Common Examples of Rays in Real-Life Geometry?

Examples of rays in real-life geometry include sunlight rays, laser beams, and the path of a moving vehicle’s headlights on the road.

Can a Ray Change Direction?

No, a ray does not change direction. It extends infinitely in one direction from its origin point. It remains fixed in direction.