Segments and rays

Segments and rays are under the subset of lines. A segment is a part of a line having two endpoints and has a particular length. On the other hand, a ray is also a part of line having one endpoint and the other direction extends indefinitely. Figure 1 illustrates a segment and a ray.

Segment-Ray

Terms to remember

  • Congruent

    - having the same size and shape.

  • Distance

    - a numerical value that describes how far two objects are.

  • Endpoint

    - a point located at the end part of a segment, defining its limit.

  • Opposite

    - reverse in position or direction.

  • Subset

    - a portion of a set.

  • Vertical

    - up-down position.


Definition of a Segment

A segment is a set of points consisting of two points of the line called the endpoints, and all of the points of the line between the endpoints. It is commonly used to represent the length, height, or width of a certain object and the distance between two objects. It is named by using the label of its endpoints and insert a line (( ) ̅) above the letters. Figure 2 shows segment AB which can also be written as (AB) ̅.

Segment

Example 1

Draw (CD) .

Answer:

two-point-label

Explanation:

Draw two points and label it as C and D.

Connect the two points straight each other.

Example 2

How many segments are there in the line below?

Answer : 6

line-endpoints

Explanation:

Use the points in the line as endpoints of the segments.

The segments are (QR) , (RS) , (ST) , (QS) , (RT) , and (QT) .


Midpoint of a Segment

A midpoint of a segment is the point that divides the line segment into two congruent parts. It is located at the center of the segment. Figure 3 shows the midpoint of a segment.

Midpoint

Note: The two vertical lines indicate that the distances from the midpoint to both endpoints are equal.

Example 3

The length of (XY) is 12 and Z is the midpoint of (XY) . Find the length of (XZ) .

Answer: 6

Explanation:

The midpoint Z divides (XY) into two congruent parts: (XZ) and (YZ) .

The length of (XZ) is ½ the length of (XY) which is ½ 12 = 6.

Example 4

O is the midpoint of (NP) . If (NO) = 9, what is the length of (NP) ?

Answer: 18

Explanation:

The midpoint O divides (NP) ̅ into two congruent parts: (NO) ̅ and (PO) ̅.

sThe length of (NP) ̅ is twice the length of (NO) ̅ which is 2 9 = 18.


Addition and Subtraction of Segments

Figure 4 illustrates three collinear points E, F, and G forming a segment.

Three-Collinear-Points

Points E and G constitute the endpoints of the segment which is (EG) and point F in between divides (EG) into two segments: (EF) and (FG) . The sum of the lengths of (EF) and (FG) is equal to the length of (EG) . Therefore, (EF) + (FG) = (EG) . The expression represents a segment if point F is between Points E and G.

The following expressions are also true for the lengths of the segments:

(EF) = (EG) – (FG)

(FG) = (EG) – (EF)

Example 5

Find the length of (JL) in the figure.

Points-J-L-are-endpoints

Answer: 10

Explanation:

Points J and L are endpoints of (JL) and point K is between points J and L.

Therefore, (JK) + (KL) = (JL) .

(JL) = 6 + 4 = 10.

Example 6

The length of (UV) is 13 and W is between points U and V. If (WU) = 5, what is the length of (VW) ?

Answer: 8

Points U and V are endpoints of (UV) and point W is between points U and V.

Therefore, (UW) + (VW) = (UV) .

(UW) = (WU) = 5

So, (VW) = (UV) - (UW) = 13 – 5 = 8.


Definition of a Ray

A ray consists of a point on a line and all points on one side of the point. It has only one endpoint. Rays are commonly used in physics to denote direction and also force. In naming a ray, consider two points in the ray: one is the endpoint and the other is any point in the ray. The label of the endpoint should be the first letter of the name of the ray and place a right arrow sign ( ( ) ) above the letters. Figure 5 illustrates a ray labeled as (RY) .

ray

Opposite rays are two rays that lie on the same line having a common endpoint and no other point in common. In Figure 6 shown, (RS) ̅ and (ST) ̅ are opposite rays.

Opposite-rays
Example 7

Draw (AE) .

Answer:

two-point-label

Explanation:

Draw two points and labeled it as A and E.

Connect the two points straight each other and extend the line from point E.

Example 8

Find the number of rays that can be found in the line.

Answer:

Explanation: 12

Consider two points in the line as the points of the ray.

Rays pointing to the right can be named as (MB) , (BP) , (PC) , (MP) , (BC) , and (MC) .

Rays pointing to the left can be named as (CP) ,(PB) , (BM) , (CB) , (PM) , and (CM) .

There are 12 rays that are found in the line.






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