1.3  Rays & Line Segments

Ray:

A ray is: Part of a line that starts at an endpoint* and extends infinitely in one direction. Named by its endpoint* and any other point on the ray, with an arrow on top. The arrow indicates the direction of the ray, from the endpoint through any other point on the ray. Half of a line that can have any point on a line as its endpoint*. Two rays on the same line going in opposite directions are opposite rays. Two opposites rays can form a line. *NOTE: In mathematics we call any point that identifies where the line stops at, as an endpoint.                But you are thinking that a ray is "starting" at that point so why not call it the "starting point"?                 We need to remember that a ray is PART of a line, and a line does not have a starting point                or an endpoint, so the point where the ray starts at is the endpoint of the line that the ray is on.     YES I KNOW WHAT THIS SOUNDS LIKE....

Example A

Problem:: What are the names of the ray(s), if any, that are on the following line?

Solution: Since this line has 3-different points on it we will have 4-different rays that will be on the line.  If we use point A as our starting point we will have the following ray:           Since it goes through 2-other points on the line we can have 2-different names for the same ray. If we use point B as our starting point and we will have 2-different rays, going in opposite directions: If we use point C as our starting point we will have the following ray:           Since it goes through 2-other points on the line we can have 2-different names for the same ray

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Line Segments:

A line segment is: Part of a line consisting of two endpoints and all points between them. Named by its two endpoints in either order with a straight segment drawn over them A fixed length..

Example B

Problem:: What are the names of the line segment(s), if any, that are on the following line?

Solution: Since this line has 3-different points on it we will have 3-different line segments that will be on the line as follows:

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Ruler Postulate

The Ruler Postulate states: The points on a line can be paired in a one-to-one correspondence with the real numbers such that      Any two given points can have coordinates 0 and 1.      The distance between two points is the absolute value of the difference of their coordinates.

Distance =   Note:  Distance is always positive, so absolute values is used to calculate distance

We can use The Ruler Postulate to:

• Have points on a line segment correspond with the points on a ruler.
    
• Measure the distance between two points
    
• Determine the length of a segment

   

Example C

Problem:: What is the distance between AB? What is the distance between CD? What is the distance between EF?

Solution: Since A is at 3 and B is at 8 we get the following: So the distance from A to B is 5. Since C is at -7 and B is at -4 we get the following: So the distance from C to D is 3. Since E is at -3 and B is at -2 we get the following: So the distance from E to F is 5.

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Segment Addition Postulate: 

• If B is between A and C, then AB + BC = AC.
      
• This also means that AB = AC - BC and BC = AC - AB

Example D

Problem:: Given the following, how far is it from A to C?

Solution: First we need to find the distance for AB. Then we need to find the distance for BC. Now that we know the distance for AB and BC we can add them together to find the distance of AC. If we check our work by find the distance from A to C we get: Which checks.

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Now the last example seems a bit silly to do the way we did it.  It would have been a lot quicker and easier to just find the distance from A to C to begin with.

However it is not always that simple to find the distance between two points as we will see in the next example.

Example E

Problem::	Solution: The first thing we need to do is draw out problem we were given.  Make sure that we include all of the information given to us. Now that we have the problem drawn out, we can see that the missing piece, ST is missing and that we will need to subtract RS from RT to find ST. Now we know that ST = 19	Watch the video:
Point S lies on between R and T. RS = 12 and RT = 31. What is the distance for ST?

Midpoint:

• The midpoint of a segment is the point that divides the segment into two equal parts.
   
  • So, if M is the midpoint of , then AM = MB.

Example F

Problem:: Can you find AC in terms of x, given that B is the midpoint and AB = 2x - 7?

Solution: Again, the first thing we need to do is draw out problem we were given.  Make sure that we include all of the information given to us. Since we were given the information that B was a midpoint we know that the distance between A and B will be the same as from B to C.  So we will write in 2x - 7 as the distance for BC. We can now see that we will need to add AB and BC together as follows: So we now know the distance of AC in terms of x will be      4x - 14.

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Practice Problems

Remember to show all work required to do the problem.

a. Find the distance between the points A and B.

b. Find AC in terms of x. 

c.  The drive from Newfolden  to Lancaster is about 46 miles.
     Karlstad is approximately the midpoint between Newfolden and Lancaster,
     about how far is the midpoint from either city?

Assignment:

Use the following graph to answer 1-3.

1. Find PQ.

2. Find NS.

3. Find MR.

Use the following graph to answer 4-9

4. AB =

5. BD =

6. DE =

7. DC =

 8. FD =

 9. AF =

 10. Find LN in terms of x.

11. Point G lies on between F and H.  Find GH if FG = 15 and FH = 34.

12. Point N lies on between L and M.  Find MN if LN = 2 and LM = 14.

13. Find WY in terms of n.

14. Point D lies on between C and F.    Find CD if DF = 2 and CF = 45.

15. Find the midpoint of the segment connecting points with coordinates -142 and
      53 on a number line.

16. A, B, and C are collinear, AB = 5x - 19, and BC = 3x + 4. Find an
      expression for AC if B is between A and C.

17. Points A, B, and C are collinear. Point B is between points A and C. AB = 12,
      AC = 7x + 5 and BC = 4x - 1. Find x.

18. Points D, E, and F are collinear with E between D and F. DE = 15,
      EF = x + 17, and DF = 3x - 10. Find EF and DF

19.  Suppose AB = 3x, BC = 2y + 16, AC = 60, and B is the midpoint of AC.
       Find the values of x and y.

20. The city is planning to install streetlights and wants five lights along a walkway
       of 60 yards. If there is a light at the beginning and at the end of the walkway
       and the lights are evenly spaced, what is the distance between each light?