Geometry 10 2 Arcs and Chords Goals o

Geometry 10. 2 Arcs and Chords

Goals o o Identify arcs & chords in circles Compute arc measures and angle measures 1/10/2022 Geometry 10. 2 Arcs and Chords 2

Central Angle A 1/10/2022 An angle whose vertex is the center of a circle. Geometry 10. 2 Arcs and Chords 3

Minor Arc C Part of a circle. The measure of the central T angle is less than 180. A 1/10/2022 Geometry 10. 2 Arcs and Chords 4

Semicircle C A Half of a circle. The endpoints of the arc are the endpoints of a diameter. The central angle T measures 180. D 1/10/2022 Geometry 10. 2 Arcs and Chords 5

Major Arc C A T Part of a circle. The measure of the central angle is greater than 180. D 1/10/2022 Geometry 10. 2 Arcs and Chords 6

Major Arc C BUT NOT A T D 1/10/2022 Geometry 10. 2 Arcs and Chords 7

Measuring Arcs o o An arc has the same measure as the central angle. We say, “a central angle subtendsan arc of equal measure”. A 42 42 C B Central Angle Demo 1/10/2022 Geometry 10. 2 Arcs and Chords 8

Measuring Major Arcs o The measure of an major arc is given by 360 measure of minor arc. A 42 42 D 1/10/2022 C B Geometry 10. 2 Arcs and Chords 9

Arc Addition Postulate Demonstration R T 1/10/2022 C A Geometry 10. 2 Arcs and Chords 10

What have you learned so far? o o o o o Analyze the diagram. Answer the following Answers… 1) 2) 3) 4) 5) 6) 1/10/2022 Q T 60 S P Geometry 10. 2 Arcs and Chords 40 R 120 11

Subtending Chords A O C 1/10/2022 Chord AB subtends AB. B Chord BC subtends BC. Geometry 10. 2 Arcs and Chords 12

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Theorem 10. 4 o Two minor arcs are congruent if and only if corresponding chords are congruent. 1/10/2022 Geometry 10. 2 Arcs and Chords 14

Theorem 10. 4 B A C D 1/10/2022 Geometry 10. 2 Arcs and Chords 15

Example Solve for x. 120 (5 x + 10) 5 x + 10 = 120 5 x = 110 x = 22 1/10/2022 Geometry 10. 2 Arcs and Chords 16

Theorem 10. 5 o If a diameter is perpendicular to a chord, then it bisects the chord and the subtended arc. 1/10/2022 Geometry 10. 2 Arcs and Chords 17

Example Solve for x. 52 2 x 1/10/2022 2 x = 52 x = 26 Geometry 10. 2 Arcs and Chords 18

Theorem 10. 6 o If a chord is the perpendicular bisector of another chord, then it is a diameter. Diameter 1/10/2022 Geometry 10. 2 Arcs and Chords 19

Theorem 10. 7 o Two chords are congruent if and only if they are equidistant from the center of a circle. 1/10/2022 Geometry 10. 2 Arcs and Chords 20

The red wires are the same length because they are the same distance from the center of the grate. 1/10/2022 Geometry 10. 2 Arcs and Chords 21

Example 16 Solve for x. 4 x – 2 = 16 4 x = 18 2 4 x – x = 18/4 x = 4. 5 1/10/2022 Geometry 10. 2 Arcs and Chords 22

Summary o o Chords in circles subtend major and minor arcs. Arcs have the same measure as their central angles. Congruent chords subtend congruent arcs and are equidistant from the center. If a diameter is perpendicular to a chord, then it bisects it. 1/10/2022 Geometry 10. 2 Arcs and Chords 23