Circles and Arcs Skill 46

Objective HSG-C.1/2/5: Students are responsible for finding the measures of central angles, arcs, circumference, and arc length.

Definitions In a plane, a circle is the set of all points equidistant from a given point called the center. A diameter is a segment that contains the center of a circle and has both endpoints on the circle. A radius is a segment that has one endpoint at the center and the other endpoint on the circle. Congruent Circles have congruent radii.

Definitions A central angle is an angle whose vertex is the center of the circle. One type of arc is a semicircle which is half of a circle. A minor arc is an arc that is smaller than a semicircle. A major arc an arc that is larger than a semicircle. Adjacent arcs are arcs of the same circle that have exactly one point in common.

Definitions The circumference of a circle is the distance around the circle. Coplaner circles that have the same center are concentric circles. The measure of an arc is in degrees while arc length is a fraction of a circle’s circumference. Congruent arcs are arcs that have the same measure and are in the same circle or in congruent circles.

Postulate 16: Arc Addition Postulate The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs. 𝒎 𝑨𝑩𝑪 =𝒎 𝑨𝑩 +𝒎 𝑩𝑪 Theorem 75: Circumference of a Circle The circumference of a circle is π times the diameter. 𝑪=𝝅𝒅 or 𝑪=𝟐𝝅𝒓 Theorem 76: Arc Length of a Circle The length of an arc of a circle is the product of the ratio measure of the arc to 360 and the circumference of the circle. 𝒍𝒆𝒏𝒈𝒕𝒉 𝑨𝑩 = 𝒎 𝑨𝑩 𝟑𝟔𝟎 ∗𝟐𝝅𝒓

𝑨𝑫 , 𝑪𝑬 , 𝑨𝑪 , 𝑫𝑬 𝑨𝑪𝑬 , 𝑪𝑬𝑫 , 𝑬𝑫𝑨 , 𝑫𝑨𝑪 𝑨𝑪𝑫 , 𝑪𝑬𝑨 , 𝑬𝑫𝑪 , 𝑫𝑨𝑬 Example 1; Naming Arcs a) What are the minor arcs of ⊙𝑂? O A C E D 𝑨𝑫 , 𝑪𝑬 , 𝑨𝑪 , 𝑫𝑬 b) What are the semicircles of ⊙𝑂? 𝑨𝑪𝑬 , 𝑪𝑬𝑫 , 𝑬𝑫𝑨 , 𝑫𝑨𝑪 c) What are the major arcs of ⊙𝑂? 𝑨𝑪𝑫 , 𝑪𝑬𝑨 , 𝑬𝑫𝑪 , 𝑫𝑨𝑬

Example 2; Finding Measures of Arcs What is the measure of each arc in ⊙𝑂? a) 𝐵𝐶 O C A D B 𝒎 𝑩𝑪 =𝒎∠𝑩𝑶𝑪 58ᵒ 𝒎 𝑩𝑪 =𝟑𝟐° b) 𝐵𝐷 𝒎 𝑩𝑫 =𝒎 𝑩𝑪 +𝒎 𝑪𝑫 32ᵒ 𝒎 𝑩𝑫 =𝟑𝟐+𝟓𝟖 𝒎 𝑩𝑫 =𝟗𝟎° c) 𝐴𝐵𝐶 𝑨𝑩𝑪 𝒊𝒔 𝒂 𝒔𝒆𝒎𝒊𝒄𝒊𝒓𝒄𝒍𝒆 𝑨𝑩𝑪 =𝟏𝟖𝟎° d) 𝐴𝐵 𝒎 𝑨𝑩 =𝒎 𝑨𝑩𝑪 −𝒎 𝑩𝑪 𝒎 𝑨𝑩 =𝟏𝟖𝟎−𝟑𝟐 𝒎 𝑨𝑩 =𝟏𝟒𝟖°

Example 3; Finding Distance A 2-ft-wide circular track for a camera dolly is set up for a movie scene. The two rails of the track form concentric circles the radius of the inner circle is 8 ft. How much farther does a wheel on the outer rail travel than a wheel on the inner rail of the track in one turn? 𝑪 𝒊𝒏𝒏𝒆𝒓 =𝟐𝝅𝒓 𝑪 𝒐𝒖𝒕𝒆𝒓 =𝟐𝝅𝒓 𝑪 𝒊𝒏𝒏𝒆𝒓 =𝟐 𝟖 𝝅 𝑪 𝒐𝒖𝒕𝒆𝒓 =𝟐 𝟏𝟎 𝝅 𝑪 𝒊𝒏𝒏𝒆𝒓 =𝟏𝟔𝝅 𝑪 𝒐𝒖𝒕𝒆𝒓 =𝟐𝟎𝝅 𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆=𝟐𝟎𝝅−𝟏𝟔𝝅 𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆=𝟒𝝅≈𝟏𝟐.𝟓𝟕 A wheel on the outer track travels 12.57 ft. farther than a wheel on the inner track.

Example 4; Finding Arc Length What is the length of each arc bold arc? Leave your answer in terms of pi. a) Diameter = 16 in. b) Radius = 15 cm. O Y X O X Y P 𝒍𝒆𝒏𝒈𝒕𝒉 𝑿𝒀 = 𝒎 𝑿𝒀 𝟑𝟔𝟎 ∗𝝅𝒅 𝟐𝟒𝟎° 𝒍𝒆𝒏𝒈𝒕𝒉 𝑿𝑷𝒀 = 𝒎 𝑿𝑷𝒀 𝟑𝟔𝟎 ∗𝟐𝝅𝐫 𝒍𝒆𝒏𝒈𝒕𝒉 𝑿𝒀 = 𝟗𝟎 𝟑𝟔𝟎 ∗𝟏𝟔𝝅 𝒍𝒆𝒏𝒈𝒕𝒉 𝑿𝑷𝒀 = 𝟐𝟒𝟎 𝟑𝟔𝟎 ∗𝟐 𝟏𝟓 𝝅 𝒍𝒆𝒏𝒈𝒕𝒉 𝑿𝒀 =𝟒𝝅 𝒊𝒏. 𝒍𝒆𝒏𝒈𝒕𝒉 𝑿𝑷𝒀 =𝟐𝟎𝝅 𝒄𝒎

#46: Circles and Arcs Questions? Summarize Notes Homework Quiz