Arcs – Types, Measure, and Identification

Key Concepts

• Naming arcs.
• Find measures of arcs.
• Adjacent arcs.
• Arc addition postulate
• Find measures of adjacent arcs.
• Congruent circles and arcs.
•  Identify congruent arcs.

Central angle

A central angle of a circle is an angle whose vertex is the center of the circle.

 In the diagram, ∠ BAC is a central angle of ⨀A.

Two points A and B are on the circle, and they are dividing the circumference of the circle into two parts.

The part of the circle between any two points on it is called an arc.

If m∠ BAC is less than, then the points on ⨀A. That lie in the interior of ∠ BAC forms a minor arc with endpoints B and C.

BC is called an ‘arc’, and it is denoted by.

If the endpoints of an arc become the endpoints of a diameter, then such an arc is called a semicircular arc or a semicircle.

If the arc is longer than a semicircular arc or a semicircle, then the arc is called a major arc.

Naming arcs:

Minor arcs are named by their endpoints. The minor arc associated with ∠ ACB is named CAB.

Major arcs and semicircles are named by their endpoints and a point on the arc.

The major arc associated with ∠ ACB can be named.

Measuring arcs:

The measure of a minor arc is the measure of its central angle.

The expression m   is read as “the measure of arc AB.”

The measure of the entire circle is 360°.

The measure of a major arc is the difference between 360° and the measure of the related minor arc.

The measure of a semicircle is 180°.

Example 1:

Find measures of arcs:

Find the measure of each arc, where is a diameter.

 a.                   b.                c.   

Solution:

1.   is a minor arc, so m  = m∠ RPS = 110°.
2.   is a major arc, so m  = 360° – 110° = 250°.
3.  is a diameter, so is a semicircle, and m   = 180°.

Adjacent arcs

Two arcs of the same circle are adjacent if they have a common endpoint. You can add the measures of two adjacent arcs.

Arc Addition Postulate

Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

Example 2:

Survey

A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or another person. The results are shown in the circle graph. Find the indicated arc measures.

a.                             b.                         c.                          d.  

Solution:

1.  =  +  

                = 29°+108°

                =137°

Example 2:

•  =   +

              = 137° + 83°……………from (a)  = 137°

∴     = 220°

•  =360° +

              = 360° – 137° ………….. from (a)  = 137°

∴     = 223°

•  =360° +

              = 360° – 61°

∴     = 229°

Guided Practice for Examples 1 and 2

Identify the given arc as a major arc, minor arc, or semicircle, and find the measure of the arc.

1.                   2.              3.            4.                5.                6.     

Solution:

1. m = 120° minor arc
2. m  =m +m  ………. two adjacent arcs

                      = 180°+ 60°………. =semicircular arc

∴m   = 240° major arc

• m   = m + m  ……… two adjacent arcs

              = 120° + 60°

∴m   = 180° semicircular arc

• m  =m  – m  ……….

              = 180°- 80°………. =semicircle arc

∴m   = 100°    minor arc

• m  = 80°
• m =180°……… = semicircle arc

Congruent circles and arcs

Two circles are congruent circles if they have the same radius.

 Two arcs are congruent arcs if they have the same measure and they are arcs of the same circle or of congruent circles.

If ⨀C is congruent to ⨀D, then you can write ⨀C ≅ ⨀D.

Circles with the same radius are congruent.

Circles with the same measure of arcs are congruent.

Identify congruency arcs. 

Tell whether the red arcs are congruent. Explain why or why not.

Solution:

1.  ≅  because they are in the same circle and  ≅ m  =80°.
2. and have the same measure but are not congruent because they are arcs of circles that are not congruent.
3.  ≅  because they are in congruent circles and  ≅  =95°.

Guided Practice for Example 3

Tell whether the red arcs are congruent. Explain why or why not.

Solution:

1.m = m =145°

Both circles have the same measures.

The radius of both circles is equal.

So, these circles are congruent.

2.m = 120°

m = 120°

Both circles have the same measures.

But the radius of the given circles is not equal.

So, the given circles are not congruent.

1. A clock with hour and minute hands is set to 1:00 P.M.
2. After 20 minutes, what will be the measure of the minor arc formed by the hour and minute hands?
3. At what time before 2:00 P.M., to the nearest minute, will the hour and minute hands form a diameter?

Solution:

1. The angle measured between any two consecutive numbers on a clock =

Time after 20 minutes =1+20=1:20 hrs.

                                              =1  hrs.

The measure of the minor arc formed by the hour and minute hands = time x angle

• At what time before 2:00 P.M., to the nearest minute, will the hour and minute hands form a diameter?

Solution:

Let’s adjust the hour and minute hands to form a diameter.

By observing the clock, the time =1:38 hrs

                                                      =1  

                                                       =1  hrs

The measure of the minor arc formed by the hour and minute hands.

The angle between the 2 and the 7 =5 x =

The remaining angle=   ×  =  

So, the angle between the hands of a clock at 1:38= + =

The nearest time to form a diameter is 1:38 hrs. before 2:00 P.M.

Problem-solving

• The deck of a bascule bridge creates an arc when it is moved from the closed position to the open position. Find the measure of the arc.

Solution:

The measure of the arc=

Let’s Check Your Knowledge

1.  and are diameters of F. Determine whether the arc is a minor arc, a major arc, or a semicircle of ⨀F. Then find the measure of the arc.

i).                         ii).                               iii).                   iv).  

• Tell whether the red arcs are congruent. Explain why or why not.
• Two diameters of ⨀P are and. If m  = 20°, find m  and m.
• ⨀P has a radius of 3 and has a measure of 90°. What is the length of it?

A .3                        B .3                C. 6                     D. 9

• In ⨀R, m =60°, m =25°, m =70°and m =20°. Find two possible values for m

Answers:

1.  
2. m   =70°
3. m  =m  – ( +  )

              =180° – (45°+70°)

              =180° -115°

              =65°

1. m   =m  +m

                 = 180° +65°

                 = 245°

1. m  = m  + m 

               = 180° +70°

               = 250°

• i)    m  = m   -( +  )

             =180°-110°

             =70°

m =m    

≅   because they are in the same circle and m =m   

2. ii)     m  = m   =85°

They are concentric circles. So, they are not in the same radius.

  and have the same measure but are not congruent because they are arcs of circles that are not congruent.

2. iii)    Radius of first circle =8 Units

              Radius of second circle = = =8 Units

              The circles are with the same radius. So, they are congruent.

              m  = m   =92°

                   because they are in congruent circles and m  = m 

• Two diameters of ⨀P are and. If m  = 20°, find m  and m.
• m  =360°- m  

               =360°-20°

               =340°

1. m = m  – m

           =180°-20°    ……..The CD is diameter,   is a semi-circle

           =160°

• ⨀P has a radius of 3 and a measure of 90°. What is the length of?

A .3              B .3           C.6                D. 9

Solution:

= 90°

∆APB is a right-angled triangle.

Radius =AP=BP=3 Units

AB=Hypotenuse

By the Pythagorean theorem

= +

        = +

         =9+9

         =18

 AB =

 AB=3  Units

• In ⨀R, m =60°, m =25°, m =70°and m =20°. Find two possible values for m

Solution:

1. Minor arc

m  =m +m +m

           = 60°+25°+70°+20°

           =175°

1. Major arc

m  = 360°- 175°

           = 185°