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Learn to recognize major arcs, minor arcs, and central angles along with their measures in circles. Discover formulas to find arc length and explore vocabulary related to circles, arcs, and angles.
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Transparency 10-2 D B A C 5-Minute Check on Lesson 10-1 • Refer to ⊙F. • Name a radius • Name a chord • Name a diameter • Refer to the figure and find each measure • 4. BC • 5. DE • 6. Which segment in ⊙C is a diameter? FL, FM, FN, FO LN, MO, MN, LO LN, MO 3 13 Standardized Test Practice: A B C D D AB AC CD CB Click the mouse button or press the Space Bar to display the answers.
Lesson 10-2 Angles and Arcs
Objectives • Recognize major arcs, minor arcs, semicircles, and central angles and their measures • central angles sum to 360° • major arcs measure > 180° • minor arcs measure < 180° • semi-circles measure = 180° • Find arc length • Formula: C • (central angle / 360°) % of circle that is the arc
Vocabulary • Central Angle – has the center of the circle as its vertex and two radii as sides • Arc – edge of the circle defined by a central angle • Minor Arc – an arc with the central angle less than 180° in measurement • Major Arc – an arc with the central angle greater than 180° in measurement • Semicircle – an arc with the central angle equal to 180° in measurement • Arc Length – part of the circumference of the circle corresponding to the arc
y x Circles - Arcs Semi-CircleEHF Major Arc BEG E Diameter (d) Center CentralAngle B F BHG G Minor Arc H
y x Circles - Probability Pie Charts Probability0 = no chance1 = for sure 90° 135° 135º------ = 3/8 360º or .375 or 37.5% Diameter (d) Radius (r) 0° 180° Center 45º------ = 1/8 360º or .125 or 12.5% 180º------ = 1/2 360º or .5 or 50% 315° 270° Circumference = 2πr = dπ
Find . Example 2-1a EXAMPLE 1
The sum of the measures of Use the value of x to find Example 2-1b (CONT) Substitution Simplify. Add 2 to each side. Divide each side by 26. Given Substitution Answer: 52
ALGEBRA Refer to . Find . EXAMPLE 2 Example 2-1c
form a linear pair. (CONT) Linear pairs are supplementary. Substitution Simplify. Subtract 140 from each side. Answer: 40
ALGEBRA AD and BE are diameters a. Find m b. Find m Example 2-1e EXAMPLE 3 Answer: 65 Answer: 40
In bisects and is a minor arc, so is a semicircle. Find . EXAMPLE 4 Example 2-2a Answer: 90
In bisects and since bisects . Find . is a semicircle. EXAMPLE 5 Example 2-2c Answer: 67
In bisects and Find . EXAMPLE 6 Example 2-2e Answer: 316
In and are diameters, and bisects Find each measure. a. b. c. Example 2-2g EXAMPLE 7 Answer: 54 Answer: 72 Answer: 234
degree measure of arc In and . a) Find the length of . b) Find the length of arc DC. arc length circumference degree measure of whole circle In and . Write a proportion to compare each part to its whole. EXAMPLE 8 Example 2-4a
Now solve the proportion for . Multiply each side by 9 . Answer: The length of is units or about 3.14 units. (CONT) Example 2-4b Simplify. C ∙ (% of the circle) = 9π ∙ (140/360) = 7π/2 Answer: The length of arc DC is 7π/2 units or about 11 units.
Summary & Homework • Summary: • Sum of measures of central angles of a circle with no interior points in common is 360° • Measure of each arc is related to the measure of its central angle • Length of an arc is proportional to the length of the circumference • Homework: • pg 533-534; 14-23; 24-29; 32-35