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9.1 Exploring Circles 9.2 Angles and Arcs

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## 9.1 Exploring Circles 9.2 Angles and Arcs

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**9.1 Exploring Circles9.2 Angles and Arcs**Objectives: To identify and use parts of circles and to solve problems involving the circumference of a circle and To recognize major arcs, minor arcs, semicircles, and central angles To find measures of arcs and central angles To solve problems by making circle graphs.**DEFINITIONS**• circle– • circumference– • diameter – • radius–**Definition**• Chord –**DEFINITIONS (Cont)**• Minor arc– • Major arc – • Semicircle – • Adjacent Arcs –**DEFINITIONS (Cont)**• sector – • central angle – • Concentric circles –**CIRCUMFERENCE**• The circumference of a circle is equal to times the diameter or 2 times the radius C = d or C = 2r • Find the circumference of a circle whose diameter is 3 ft. Round your answer to three significant digits**Y**110 110 O Z Arc Measure The measure of a center angle is equal to the measure of the intercepted arc. Center Angle Intercepted Arc Give is the diameter, find the value of x and y and z in the figure. Example:**Find the measure of each arc.**• BC = • BD = • ABC = • AB =**Ex. 1: Finding Measures of Arcs**• Find the measure of each arc of R. 80°**Note:**• Two arcs of the same circle are adjacent if they intersect at exactly one point. You can add the measures of adjacent areas. • Arc Addition Postulate. The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. m = m + m**Ex. 2: Finding Measures of Arcs**• Find the measure of each arc. 40° 80° 110°**ARC LENGTH**• The length of an arc equals the ratio of the number of degrees of the arc to 360° times the circumference • Determine the length of a 30° arc on a circle with a radius of 5 m:**POSTULATE EXAMPLE**Determine the length of arc AB in the figure below given that CD is 24 cm, COD = 92, and AOB = 58 C D O A B 14**PRACTICE PROBLEMS (Cont)**Determine the circumference of a circle with a radius of 2.5 inches. Determine the arc length of a circle with a 5 m radius and a 50° arc. 15**Congruent Arcs**• Two arcs which have the same measure are not necessarily congruent arcs. • Congruent arcs have the same measure and are parts of the same circle or congruent circles.**Relating Congruent Arcs, Chords & Central Angles**• Theorem: If two angles of a circle (or of congruent circles) are congruent, then their intercepted arcs are congruent. • Theorem: If two arcs are congruent, then their central angles are congruent.**Circles that have a common center are called concentric**circles. Concentric circles No points of intersection Concentric circles**Here is a circle graph that shows how people really spend**their time. Find the measure of each central angle in degrees. • Sleep • Food • Work • Must Do • Entertainment • Other