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Lesson 2 Rays, Segments,  and Length

Page history last edited by Math in a Box - Susan Johnsey gm 4 years, 9 months ago

Lesson 2    Rays, Segments, and Length               Susan O. Johnsey

                                                                                                                       www.mathinabox.com

                                                                                                                       www.mygeometryreview.pbworks.com

 

DID YOU LOOK FOR MY EMAILS for your LESSON 1 NOTES?   Please do that now if you have not.

 

  I will continue to give you new words to understand. 

     In your STUDENT folder did you find your Geometry Collection?

          You really need to use it.  That collection will make this class much easier.

 

SEGMENTS are part of a line.
Each segment has 2 ENDPOINTS; we usually label them with a capital letter.

           A_____________________________B   This is segment AB or segment BA.

           Some times they are called line segments, but most teachers and books shorten it to segments.

     The segments do not go on forever as the lines doWe can measure the length of a segment.

   

      Some books write a short line (segment) above the AB.

 

ALL YOU NEED TO KNOW ABOUT RAYS:

 

Look at the LINE in the GREEN box below first.   All my comments are about the rays in that box. 

 

Then be sure you understand the info in the yellow and blue boxes.

A yellow box is below this sentence if you do not see it let me know. Important to see diagrams in online classes.

 

Do you see also that ray CB and ray CA are the same ray?

 

       They start at the same point C and then go in the same direction!

 

When you write the name of a ray be sure that your first letter is the point where the ray begins.  

Ray CB begins at point C.  

Ray BC begins at point  B.  These are not opposite rays nor same rays.  

 

Do you see also that ray CD and ray CA are opposite rays?  

 

They start at same point C and then go in opposite directions!

 

 

I hope you are taking some good notes. 

There are some quizzes and games at the end of this lesson.

 

Diagram  <--W----------X-------------Y-----Z----->   Please pretend this is a line with 4 labeled points!

Answers can be seen by highlighting the brown area. 

 

1.  Give 3 names for this line.      

     WX  or WY  or WZ or XY or XZ or YZ  then draw a tiny line (with its arrows) above the 2 letters.

 

2.  Name 3 segments in the line.     WX  or WY  or WZ or XY or XZ or YZ  then draw a tiny line segment above the 2 letters. 

 

3.  Name two rays that are opposite rays.  XW and XY then draw a tiny ray above the two letters.

                                                                     YW and YZ then draw a tiny ray above the two letters.                                  

 

4.  Name two rays that are the same ray.    YW and YX then draw a tiny ray above the two letters.  

                                                                     ZW and ZY then draw a tiny ray above the two letters.

                                                                     WX and WY then draw a tiny ray above the two letters. And there are others.

 

BE very sure when naming SAME rays and OPPOSITE rays that you write the same letter first. 

  1.   Ray XZ is not the same as ray ZX and they are not opposite rays either!  
  2.   Ray XZ begins at X and goes in the direction toward and through Z
  3.   The first letter given for a RAY is important; it must be where the ray begins.  Ray ZX begins at Z and goes in the direction toward and through X.

 


 

MORE special words to know about SEGMENTS:

 

     Congruent Segments   (this is easy):  line segments that have the same or equal lengths.

 

       Midpoint:   a point that divides the  segment into two congruent (equal length) segments

              The midpoint is the point in the middle of the segment.

 

     Bisector of a segment  can be a ray or line or segment that passes through the midpoint of a segment.

 

HOW many items do you have in your Geometry Collection?

 

           Watch this animation of constructing a segment bisector with a compass and ruler!  CLICK here.

 

  YOU SHOULD SEE A DRAWING OF SEGMENT AB  BELOW.     YOU MUST BE ABLE TO VIEW THE DIAGRAMS THAT I HAVE IN  THE LESSONS.   LET ME KNOW IF YOU CAN NOT SEE THE DRAWING BELOW. 

    M is the midpoint of AB.    SO WHAT DO WE KNOW? 

  the length of segment AM equals the length of segment MB 

      the length of segment AM = the length of segment MB .

We know this by the Definition of Midpoint.


 

Now draw a line or segment or ray through midpoint M and it is called a BISECTOR.

 

Some books use the letter m to mean measure so mAB also means the distance from A to B.

You can think of it, mAB,  as measuring the LENGTH of segment AB.

 

EXAMPLE of using algebra with your geometry

 

GIVEN:   A is the midpoint of BC.  Please draw segment BC and its midpoint A and label it.   

NOTE that midpoint A is between B and C and these are collinear points.

 

  Let me also tell you that    BA = 3x+6  and AC = 2x +14 .

My Question:  What is the measure of AC ?                                                                           

 

Because A is the midpoint, you know that BA = AC by definition of midpoint.  


NOW  here is how we use ALGEBRA  and GEOMETRY together.

 

             I have written the algebra on the left and the geometry on the right.

 

ALGEBRA                   GEOMETRY reasons  are in GREEN

      BA = AC               Definition of midpoint. GEOMETRY: remember A is midpoint of BC.

3x + 6 = 2x + 14       You are given : BA = 3x + 6 and AC = 2x + 14

                                                 so in algebra column replace the BA with 3x+6

                                                 and the AC with 2x+14.  We call this substitution property.

     3x  = 2x + 8           to get this line I subtracted 6 from each side.    

                                     That is the subtraction property.

        x= 8                     I subtract 2x from each side.  

                                     I hope you know your algebra.

                                    That is the subtraction property.

 

Now substitute 8 for x in the expression 2x + 14.

AC = 2x + 14                 Original measure;  it was given to us.

AC = 2(8) + 14              Substitute  x = 8.   Substitution property

AC = 16 + 14 = 30       Simplify.

     BA = AC =30    so the segment BC= 60.

 

Want another Example using Algebra and Geometry?

Algebra Example using the Segment Addition Postulate   

 

We have 3 collinear points:  M and B and N.  Please pretend there is segment MB and BN below.

 You are given measure of segment MN is 24 inches and this diagram:

 

   ........................................24.........................

   .________  ._________________.

 M       x       B                x +6       N

 

The length of BN is 6 more than the length of MB.   Hope you see the x and x+6 are the lengths of MB and the BN.

See the 3 segments MB   and  BN   and MB.

Do you know why segments MB + BN = segment MN?

The REASON in geometry is the Segment Addition Postulate.

All THREE points must be on ONE line segment (Collinear) to use the SEGMENT ADDITION postulate.

 

 The  "parts of a segment added together = the whole segment" .

 Be sure you know Segment Addition Postulate.  Is it in your Collection?

 

We will replace (substitute) the MB with x.   See the diagram

And replace the BN with x+6.  BN is 6 longer than MB.    WATCH below.

 

  Algebra                                      Geometry

  MB + B N = MN       from the Segment Addition Postulate.   This becomes

 

   x   x+6 =  24     WE replaced the  MN with 24, that was

                                  given in diagram above too.  Did you see that?  

                                                   We call that substitution property in geometry.      

Now solve.

 

   2x + 6    = 24             YES you must show these steps.  

      2x    = 24 - 6           Guessing the numbers is NOT helpful.

          2x =18

         so x =9.    

Recall in the diagram that   x   is MB so MB = 9  and BN = x+6  so BN = 9+6 = 15.

 

Can you find the length of the segments?  no algebra needed here. 

  Draw the segment and the points.  You will use add, subtract or divide!

 

 A_________________B__________C__________D__________________E

 

Pretend the above is segment AE with 5 points labeled on it.  The above is NOT drawn to scale! 

There are infinitely many points on a line; we label a few of them.

You cannot measure it, nor use a ruler.  You must use the information that I give you to decide the measurements.

 

Let AB= 9 cm and let AE = 22 cm  and also know that AB=ED.   How long is segment BD?    22-9 -9 = 4 cm.

How long is BC if BC=CD?    Oh do you see that means C must be midpoint of BD?   Thus BC is half of BD.   BC = 2cm.

Highlight the brown boxes so you can see the answers.

 

You will need to know one more definition!

         "Co-or-di-nates" are what we call the numbers on our number lines.

 You will see some number lines in the problems below.   

Write the word Coordinates in your Geometry Collection.


Here is a math site that would like you to join, but do not do that.   Take notes and  write your answers.

  1. Just click on the GEOMETRY class on that page and then you can complete the FREE Practice  questions.  
  2. On the Geometry page for segments and rays follow the numbered steps on left side of page.  YOu can click  INSTRUCTION or Practice or BONUS or TEST.  Choose the one you want to do and then begin by clicking the 1.  First click INSTRUCTION and then choose a number. The NUMBERed steps give you videos to watch if you do not understand well the rays, segments, and length of segments.  OR click Practice or Bonus or TEST.   For each one you then choose the numbers and follow the directions.    I want you to do TEST.  If you are ready then click TEST and then read the first question.
  3. When click TEST then you will see the first question on the right side. READ it.
  4. Click ANSWERS Choices when you are ready to answer the question.  
  5. Then choose A, B, C , or D and you will be told if that is correct or not. 

Complete ALL the TEST QUESTIONS. So click number 2 now and finish them all.   This is for you to practice the SEGMENTS and RAYS skills.   I want you to really work hard on these, and if you find one that you just cannot figure out then please SEND IT TO ME.   You can take a screenshot.

       You must choose reset and complete 5 more questions if you miss 1 out of 5.  If you miss none then you can stop if you wish.

 

                  PRACTICE QUESTIONS for SEGMENTS AND RAYS

 

HERE are 2 GAMES THAT WILL HELP YOU know how well you know the CONCEPTS.  

PLAY them MORE THAN ONCE;  PLAY  them  UNTIL  YOU CAN earn the $1 million dollars.  

 I am the member at the game web site, QUIA;  you do NOT need to join.

 

Write down  HOW LONG THAT IT TAKES YOU to win the 1 million dollars.  

 

Please play these until you win.  If you cannot then please ask me questions.

 

Lines, Segments, Rays   http://www.quia.com/rr/356548.html

RECALL:       "Co-or-di-nates" are what we call the numbers on our number lines.

 

And another game to help find the measurement (length) of segments:  

Segment Measures  http://www.quia.com/rr/852100.html 

 

 

HOW did you do on the activities above?  ____

What did you learn? ____

Completer YOUR LESSON 2 NOTES  web page.

Look for it in your STUDENT folder.

 

Questions?     mathinabox@gmail.com                                   www.mathinabox.com                              Susan O. Johnsey  2015