Geometry Arcs and Chords

September 13, 2015 Goals  Identify arcs & chords in circles  Compute arc measures and angle measures

September 13, 2015 Central Angle An angle whose vertex is the center of a circle. A

September 13, 2015 Minor Arc Part of a circle. The measure of the central angle is less than 180. A C T

September 13, 2015 Semicircle Half of a circle. The endpoints of the arc are the endpoints of a diameter. The central angle measures 180. A C T D

September 13, 2015 Major Arc Part of a circle. The measure of the central angle is greater than 180. A C T D

September 13, 2015 Major Arc BUT NOT A C T D

September 13, 2015 Measuring Arcs  An arc has the same measure as the central angle.  We say, “a central angle subtends an arc of equal measure”. 42 A B C Central Angle Demo

September 13, 2015 Measuring Major Arcs  The measure of an major arc is given by 360  measure of minor arc. 42 A B C D

September 13, 2015 Arc Addition Postulate R A C T Postulate Demonstration

September 13, 2015 What have you learned so far?  Page 607  Do problems 3 – 8.  Answers…  3)  4)  5)  6)  7)  8) P Q R S T 120 60 40

September 13, 2015 Subtending Chords A B O C Chord BC subtends BC. Chord AB subtends AB.

September 13, 2015

Theorem 12.4  Two minor arcs are congruent if and only if corresponding chords are congruent.

September 13, 2015 Theorem 12.4 A B C D

September 13, 2015 Example 120 (5x + 10) Solve for x. 5x + 10 = 120 5x = 110 x = 22

September 13, 2015 Theorem 12.5  If a diameter is perpendicular to a chord, then it bisects the chord and the subtended arc.

September 13, 2015 Example 52 2x Solve for x. 2x = 52 x = 26

September 13, 2015 Theorem 12.6  If a chord is the perpendicular bisector of another chord, then it is a diameter. Diameter

September 13, 2015 Theorem 12.7  Two chords are congruent if and only if they are equidistant from the center of a circle.

September 13, 2015 The red wires are the same length because they are the same distance from the center of the grate.

September 13, 2015 Example 16 4x – 2 Solve for x. 4x – 2 = 16 4x = 18 x = 18/4 x = 4.5

September 13, 2015 Summary  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.

September 13, 2015 Practice Problems