Unit 4: CIRCLES Topic 11: C IRCLE M EASUREMENTS Topic 12: T HEOREMS A BOUT C IRCLE

Topic 11: C IRCLE M EASUREMENTS In a plane, a circle is the set of all points equidistant from a given point called the center. You name a circle by its center. Circle P ( ⊙ P) is shown below. 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. A central angle is an angle whose vertex is the center of the circle Circles and Arcs

Topic 11: C IRCLE M EASUREMENTS An arc is a part of a circle. One type of arc, a semicircle, is half of a circle. A minor arc is smaller than a semicircle. A major arc is larger than a semicircle. Adjacent arcs are arcs of the same circle that have exactly one point in common. You name a minor arc by its endpoints and a major arc or a semicircle by its endpoints and another point on the arc Circles and Arcs

Topic 11: C IRCLE M EASUREMENTS The measure of a major arc is equal to the measure of the related minor arc subtracted from 360. Arc Measure The measure of a minor arc is equal to the measure of its corresponding central angle. The measure of a semicircle is 180. Postulate 11-1 Arc Addition Postulate The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs Circles and Arcs

Topic 11: C IRCLE M EASUREMENTS Naming Arcs 11-1 Circles and Arcs

Topic 11: C IRCLE M EASUREMENTS Circumference of a Circle The circumference of a circle is  times the diameter. Arc Length 11-1 Circles and Arcs

Topic 11: C IRCLE M EASUREMENTS A bike has wheels the same size as the wheel shown below. If the bike wheels are rotating at 320 revolutions per minute, what is the speed of the bike in miles per hour? Explain your reasoning. (Hint: 1 mile = 5280 ft) Evaluate Reasonableness (1)(B) Explain how you know your solution is reasonable Circles and Arcs

Topic 11: C IRCLE M EASUREMENTS a. A car has a circular turning radius of 16.1 ft. The distance between the two front tires is 4.7 ft. How much farther does a tire on the outside of the turn travel than a tire on the inside? Finding a Distance b.Suppose the radius of ⊙ A is equal to the diameter of ⊙ B. What is the ratio of the circumference of ⊙ A to the circumference of ⊙ B? Explain. circle A circle B Let r = radius of circle B 2r2r 11-1 Circles and Arcs

Topic 11: C IRCLE M EASUREMENTS What is the length of a semicircle with radius 1.3 m? Leave your answer in terms of  Circles and Arcs

Topic 11: C IRCLE M EASUREMENTS 2. A piece of plywood was cut out in the shape below for scenery in a school play. The curved side consists of four 90  arcs with the same radius. What is the perimeter of the shape? Round to the nearest tenth of a foot. 3. A 30  arc of ⊙ P has the same length as a 36  arc of ⊙ Q. What is the ratio of the radius of ⊙ P to the radius of ⊙ Q ? ANSWER: 6 : Circles and Arcs

Topic 11: C IRCLE M EASUREMENTS 11-1 Circles and Arcs Do you understand? 4. Vocabulary What is the difference between the measure of an arc and arc length? Explain. The measure of an arc corresponds to the measure of a central angle; the measure of an arc length is a fraction of the circle’s circumference. 6. Analyze Mathematical Relationships (1)(F) The circumference of ⊙ A is twice the circumference of ⊙ B. How is the radius of ⊙ A related to the radius of ⊙ B? Explain. The radius of ⊙A is twice radius of ⊙B.