Arcs and Central Angles Section 11.3

Goal Use properties of arcs of circles.

Key Vocabulary Central angle Arc Minor Arc Major Arc Semicircle Congruent Circle Congruent Arcs Adjacent Arcs Arc Length

Postulates 16 Arc Addition Postulate

Central Angle (of a circle) Central Angle (of a circle) NOT A Central Angle (of a circle) Central Angle An angle whose vertex lies on the center of the circle. Definition:

Central Angle Central angle- Its sides contain two radii of the circle. A B C  ACB is a central angle Radius

Sum of Central Angles The sum of the central angles of a circle = 360 o –as long as they don’t overlap.

Example 1

Refer to ⊙T. Assume RV is a diameter. Find the m ∠QTR. Example 1b:

Linear pairs are supplementary. Substitution Simplify. Subtract 140 from each side. form a linear pair. Answer: 40 Example 1b: ∠ QTR and ∠ QTV m ∠ QTR + m ∠ QTV = 180 m ∠ QTR + 20x = 180 m ∠ QTR + 20(7) = 180 m ∠ QTR = 180 m ∠ QTR = 40

Answer: 65 Answer: 40 Refer to ⊙Z. Assume AD and BE are diameters. a. Find m ∠ CZD b. Find m ∠ BZC Your Turn:

O Centre An arc is the distance between any two points on the circumference of a circle. Symbol: K L ⏜

Every central angle cuts the circle into two arcs The smaller arc is called the Minor Arc. The MINOR ARC is always less than 180°. It is named by only two letters with an arc over them as in our example,. The Minor Arc The larger arc is called the Major Arc. The MAJOR ARC is always more than 180°. It is named by three letters with an arc over them as in our example,. The Major Arc Arcs

O Centre L K Y An arc divides the circle into two parts: the smaller arc is called the minor arc, the larger one is called the major arc. Minor Arc KL Major Arc KYL Minor arcs are named by their endpoints. Major arcs are named by their endpoints and another point on the arc that lies between the endpoints.

The Semicircle (Major Arc = Minor Arc) : The measure of the semicircle is 180°. SEMICIRCLES are congruent arcs formed when the diameter of a circle separates the circles into two arcs. Semicircle

 Half of a circle is called a semicircle. Centre O Diameter D E S  A semicircle is also an arc of the circle.  A semicircle is named with 3 letters, same as a major arc. R Arc DSE Semicircle DRE Semicircle DSE Arc DRE

NAME THE ARC

ARCS The part or portion on the circle from some point B to C is called an arc. A B C Arcs : Semicircle: An arc that is equal to 180°. Example: O A B C

Minor Arc & Major Arc Minor Arc : A minor arc is an arc that is less than 180° A minor arc is named using its endpoints with an “arc” above. A B Example: Major Arc: A major arc is an arc that is greater than 180°. A major arc is named using its endpoints along with another point on the arc (in order). A B C Example: O

Example: ARCS Identify a minor arc, a major arc, and a semicircle, given that is a diameter. A C D E F Minor Arc: Major Arc: Semicircle:

True or False The name of the orange arc below is…..

True or False FALSE…it is Semicircle - named using three points on the arc; endpoints listed first and last.

True or False The name of the orange arc below is…..

True or False FALSE. It is….. Minor arc - named using the two endpoint letters.

True or False The name of the orange arc below is…..

True or False FALSE Major arc - named using three points on the arc with endpoints listed first and last.

Arc Measure The measure of the an arc is equal to the measure of the central angle. An arc is measured in degrees, the same as an angle. Arc Central Angle Y Z O 110 

The measure of a minor arc is the measure of its central angle. Central Angle = Minor Arc The measure of a major arc is 360° minus the measure of its central angle. Definition of Arc Measure

Example 2 Name the red arc and identify the type of arc. Then find its measure. b. a. SOLUTION a. is a minor arc. Its measure is 40°. DF b. is a major arc. Its measure is 360° – 110° = 250°. LMN

Example 3 230° = =40° + 80° + 110° Find the measure of GEF. SOLUTION =mGH++ mGEF mHE mEF

Find the arc measures 80  45  180  125  55  305  m DFB = m AB = m DE = m AF = m DF = m BF = m BD = m FE = 135  45  55 

Example 4a Answer:

Example 4b

Answer:

Example 4c Answer:

Your Turn: A. B. C. D.

Your Turn: A. B. C. D.

Your Turn: A. B. C. D.

Within a circle or congruent circles, congruent arcs are two arcs that have the same measure. In the figure ST  UV. Congruent Arcs

Identify Congruent Arcs Example 5 Find the measures of the blue arcs. Are the arcs congruent? a.b. SOLUTION a. Notice that and are in the same circle. Because = = = 45°,. DC  mAB mDC AB DC b. Notice that and are not in the same circle or in congruent circles. Therefore, although XYZW = = mXY mZW XY  65°, ZW.

Checkpoint Identify Congruent Arcs Find the measures of the arcs. Are the arcs congruent? and BCEF 1. ANSWER mBC= 58° ; mEF= 58° ; yes and BCCD 2. ANSWER mBC= 58° ; mCD= 72° ; no and CDDE 3. ANSWER mCD= 72° ; mDE= 72° ; yes 4. and CBFBFE ANSWER mCBF 158° ; yes mBFE ==

Example 6a m  LPK= 0.21(360)Find 21% of 360. = 75.6Simplify. Answer:

Example 6b

Sum of arc in circle is 360. Substitution Simplify. Answer:

Your Turn: A B C D.165.9

Your Turn: A B C D.201.4

Adjacent arcs are arcs of the same circle that intersect at exactly one point (share an endpoint). RS and ST are adjacent arcs. Adjacent Arcs

The measure of an arc formed by two adjacent arcs is the sum of the measures of the 2 arcs. m DA = 72  m CA + m DC = 72  Postulate 16 Arc Addition Postulate mDC = 32  mCA = 40  B D A C

Example 7a Find m  ABD m  ABD = 48  m CA + m DC = m AD = m  ABD m AC = 4x + 7  m CD = 2x + 5  B D A C 8x  4x x + 5 = 8x 6x + 12 = 8x 12 = 2x 6 = x m  ABD = 8(6)

Line segments AC and BE are diameters of ⊙ F. mCFD = 180 – (97.4 + 52) = 30.6 = 97.4  = 128 mBD = mBC + mCD mBC = 97.4 Vert.  s Thm. ∆ Sum Thm. m  CFD = 30.6  Arc Add. Post. Substitute. Simplify. Find mBD. mCD = 30.6 Example 7b

Find each measure. mJKL mKPL = 180° – ( )° = 25° + 115° mKL = 115° mJKL = mJK + mKL = 140° Arc Add. Post. Substitute. Simplify. Your Turn:

Find each measure. mLJN = 295° mLJN = 360° – ( )° Your Turn:

Arc Length Another way to measure an arc is by its length. The arc length is different from the degree measure of an arc. Suppose a circle was made of string. The length of the arc would be the linear distance of that piece of string representing the arc. An arc is part of a circle, so its length is part of the circumference. We use proportions to solve for the arc length, l. degree measure of arc = arc length degree measure of circumference

Arc Length Arc length is a part of the circumference of a circle. OR C B A ℓ r x˚x˚

Example 8 Arc Length Equation Substitution Answer: Use a calculator. cm

Example 9 Arc Length Equation Substitution Use a calculator. Answer:

Your Turn: A.3.56 cm B.3.77 cm C.3.98 cm D.4.21 cm

Example 10 Find the length of the red arc. a.b. c. SOLUTION = a. Arc length of AB 50° 360° · 2  (5) ≈ 4.36 centimeters Arc length of CD 50° 360° = · 2  (7) ≈ 6.11 centimeters b. = Arc length of EF 98° 360° · 2  (7) ≈ centimeters c.

Checkpoint Find Arc Lengths Find the length of the red arc. Round your answer to the nearest hundredth ANSWER 4.19 in. ANSWER ft ANSWER 9.42 cm

Example 11 Arc Length Equation Substitution Use a calculator. Answer:

Your Turn: A cm B cm C cm D cm

Your Turn: A cm B cm C cm D cm

Assignment Pg. 604 – 607; #1 – 57 odd.