GeometryGeometry 9.3 Arcs and Chords

Geometry Geometry Objectives/Assignment Use properties of arcs of circles. Use properties of chords of circles.

Geometry Geometry Using Arcs of Circles In a plane, an angle whose vertex is the center of a circle is a central angle of the circle.

Geometry Geometry Using Arcs of Circles minor arc of the circle is less than 180 major arc of the circle is greater than 180 semicircle if the endpoints of an arc are the endpoints of a diameter.

Geometry Geometry Naming Arcs The measure of a minor arc is defined to be the measure of its central angle. 60 ° 180 ° m = m  GHF = 60 °.

Geometry Geometry Naming Arcs The measure of a major arc is defined as the difference between 360° and the measure of its associated minor arc. For example, m = 360° - 60° = 300°. The measure of the whole circle is 360°. 60 ° 180 °

Geometry Geometry Finding Measures of Arcs Find the measure of each arc of R. a. b. c. 80 °

Geometry Geometry Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. Solution: is a minor arc, so m = m  MRN = 80 ° 80 °

Geometry Geometry Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. Solution: is a major arc, so m = 360 ° – 80 ° = 280 ° 80 °

Geometry Geometry Ex. 1: Finding Measures of Arcs Find the measure of each arc of R. a. b. c. Solution: is a semicircle, so m = 180 ° 80 °

Geometry Geometry Theorem 9.1 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.  if and only if 

Geometry Geometry Theorem 9.2 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. , 

Geometry Geometry Theorem If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. is a diameter of the circle.

Geometry Geometry Using thisTheorem You can use Theorem 9.3 to find m. Because AD  DC, and . So, m = m 2x = x + 40Substitute x = 40 Subtract x from each side. 2x ° (x + 40) ° D

Geometry Geometry In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. AB  CD if and only if EF  EG. Theorem 9.3

Geometry Geometry Using Theorem 9.4 AB = 8; DE = 8, and CD = 5. Find CF.

Geometry Geometry Using Theorem 9.4 Because AB and DE are congruent chords, they are equidistant from the center. So CF  CG. To find CG, first find DG. CG  DE, so CG bisects DE. Because DE = 8, DG = =4.

Geometry Geometry Using Theorem 9.4 Then use DG to find CG. DG = 4 and CD = 5, so ∆CGD is right triangle. So CG = 3 CF = CG = 3

Geometry Geometry Practice 360 – 120 = /3 = 80

Geometry Geometry Practice 12 * 2 = 24

Geometry Geometry Practice BY² = 20² - 12² BY = 16 XY = 16*2 = 32

Geometry Geometry BC = 360 / 5 = 72

Geometry Geometry MX = 12 XN = 12 MN = PQ YQ = 12

Geometry Geometry Radius² = 8² + 6² R² = 100 R = 10 inches 16 6

Geometry Geometry Chord ² =13² - 5² Chord ² = 144 C= C= 24 inches 13 5

Geometry Geometry Using Pythagorean Formula Distance ² = 17 ² - 15 ² D ² = 64 Distance = 8 inches 17 30

Geometry Geometry Using Pythagorean Formula Radius = 6 inches  APB = 120° PAB isosceles triangle  A= (180 ° °)/2  A = 30 ° Cos 30 = x/ 6 X = 6*Cos 30 = 5.2 AB = = 10.4

Geometry Geometry PD = PC = PB = Radius CE = 2 EP = 5 -2 = 3 Using Pythagorean Formula for EPB EB ² = PB ² - PE ² EB ² = 5 ² - 3 ² EB ²= 16 EB = 4 AB = AE + EB = 4+ 4 = 8