4.2  Arcs & Angles

 

Arcs:

An arc is a part of a circle consisting of two points on the circle, called endpoints, and all the points on the circle between them.

Properties of an Arc:

Major Arc:

  • A major arc is an arc that is larger than half a circle.
        
  • The measure of a major arc must be greater than 180°and less than 360° and is the difference between 360° and the measure of the associated minor arc.
        
  • All major arcs are named using the two endpoints of the arc and a point on the circle between the endpoints.
      
    So hereis the major arc, we could also name this as.

Minor Arc:

  • A minor arc is an arc that is smaller than half a circle.
      
  • The measure of a minor arc is the same as the measure of its central angle. The measure of a minor arc must be greater than 0° and less than 180°.
        
  • All minor arcs are named using the two endpoints of the arc.
      
    So hereis the minor arc

Semicircle:

  • A semicircle is an arc equal to half a circle.
      
  • The measure of a semicircle is 180°.
      
  • Like major arcs, semicircles can be named with the two endpoints of  the semicircle and a point on the circle between the endpoints.
        
    So here we have 2-semicicles, &.

Arc Measurements:

Any arc can be measured in 2-different ways.  It can be measured by degrees or it can be measured as part ( fraction, decimal. or percent ) of the circumference called arc length..

Central Angle:

  • On a circle, the central angle is formed by intersection of any 2-radii of a circle.
      
    So here we seeis a central angle formed by the intersection of &.
        
  • The measure of an central angle is equal to twice the measure of the inscribed angle that opens to the same arc
      
    Here we see mCAB = 130°, so then mCXB  = 65°
      

Inscribed Angle:

  • An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle
      
    So here we see is an inscribed formed by the intersection of chords &.
        
  • The measure of an inscribed angle is equal to half the measure of the central angle that opens to the same arc
      
    Here we see  mCXB  = 65°, so then mCAB = 130°

Arc Measure:

  • The measure of an arc in degrees is:
        
    • Equal to the measure of the Central Angle that it opens to.

      So here we see mCAB = 130°, so the measure of
      = 130°.
          
      and/or
          
     
    • Equal to twice the measure of the Inscribed Angle that it opens to.

      So here we see mCXB = 65°, so the measure of
      = 2( 65° ) = 130°.

Arc Length:

  • Arc Length =
      
    So here the arc length of

 

Example A

Task:

Use Geogebra to create A, with radius AB = 4,
Major Arc , Minor Arc


 

 Solution:

Step:
1.  Plot Point A
2.  Create Circle: Center A & Radius 4
3.  Create Point B, anywhere on the circle.
4.  Create Point C, anywhere on the circle.
5.  Create Diameter CD
6.  Create Points E & F, on the opposite side of     
     Diameter CD from Point B.
 7.  Hide Circle A  ( or use a dotted line )
8.  Create Circular Arc EF, click on the points in
     the following order A, E, F, you now have
     Minor Arc EF
 9.  Create Circular Arc EBF, click on the points in
     the following order A, F, E, you now have
     Major Arc EBF
    
     

You now have A, with radius AB = 4,
Major Arc , Minor Arc

Press or to run the Geogebra build simulation.








     Watch the video:
Identify the parts of A:
                                               
Solution
:

Quick Question...

Instructions:

  • Use the given "keypad" to enter your answer
  • Click on the "Stopwatch" for a 30 second timer
  • Use the slider to select the numbers of questions

To use the app:

  • Clickto begin, click for full screen mode
  • Clickto check your answer.
  • Clickto goto the next problem
  • Clickto stop the app

  

 

Example B

Problem:

Use A, to find mEAF = ?, if  mEBF = 55°,

                   

                


Watch the video:
Solution:
         

 

Example C

Problem:

Use A, to find  to find the arc measure of in degrees?

                


Watch the video:
Solution:

    

Example D

Problem:

Use A, to find  to find the arc length of   if radius AB = 4

                


Watch the video:
Solution:

    

 

Quick Question...

Instructions:

  • Use the given "keypad" to enter your answer
  • Click on the "Stopwatch" for a 30 second timer
  • Use the slider to select the numbers of questions

To use the app:

  • Clickto begin, click for full screen mode
  • Clickto check your answer.
  • Clickto goto the next problem
  • Clickto stop the app

  

 

Practice Problems

Use the figure on the right to answer the following questions:

If = 110° and the length of = 4.
  
A  What is the degree measure of = ?
  
B. What is the= ?
  
C.  What is the degree measure of = ?
   
D. What is the arc length of = ?
  
E. What is the arc length of = ?

         		 

 

Assignment:

Use the information and figures on the right to answer the following questions:


1. What is the degree measure of = ?		 Diameter = 10 & = 92°
 
2.  What is the= ?
    
3.  What is the degree measure of = ?
    
4.  What type of arc ( Major, Minor, or Semicircle )
      is the arc from point C to point E on the circle?
    
5.  What is the= ?.
    
6.  What is the arc length of = ?
    
7.  What is the arc length of = ?
 
8  What is the arc length of = ?
 
    
9. What is the = ?
  		Radius = 2, = 32°, & = 164°
10.  What is the = ?
  
11.  What is the = ?
  
12. What is the degree measure of = ?
  
13.   What is the degree measure of = ?
  
14.  What is the circumference of  T?
  
15. What is the arc length of = ?
  
16.  What is the arc length of = ?
    
17. What is the degree measure of = ?
  
= 6, //, = 55°, & = 40°
18.  What is the = ?
  
19.  What is the arc length of=?
  
20.  What is the degree measure of = ?
21.  What is the arc length of =?