6.14.1 Arcs, Chords, and Angles

6.14.1 Arcs, Chords, and Angles

• 1.
  6.14.1 Arcs andChords The student is able to (I can): • Apply properties of arcs • Apply properties of chords
• 2.
  circlecirclecirclecircle – theset of all points in a plane that are a fixed distance from a point, called the center. A circle is named by the symbol and its center. A • A
• 3.
  diameterdiameterdiameterdiameter – aline segment whose endpoints are on the circle and includes the center of the circle. radiusradiusradiusradius – a line segment which has one endpoint on the circle and the other on the center of the circle. centralcentralcentralcentral angleangleangleangle – an angle whose vertex is on the center of the circle, and whose sides intersect the circle. A • C B D CD is a diameter AB is a radius ∠BAD is a central angle
• 4.
  secantsecantsecantsecant – aline that intersects a circle at two points chordchordchordchord – a line segment whose endpoints are on the circle. (A diameter is a special kind of chord.) tangenttangenttangenttangent – a line in the same plane as a circle that intersects it at exactly one point. pointpointpointpoint ofofofof tangencytangencytangencytangency – the point where the tangent and a circle intersect. • A B m C chord secant tangent point of tangency
• 5.
  commoncommoncommoncommon tangenttangenttangenttangent –a line that is tangent to two circles. common external tangent common internal tangent
• 6.
  Theorem: If aline is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Theorem: If a line is perpendicular to a radius at a point on the circle, then it is tangent to the circle. BELine t ⊥ Line t tangent to ⊙B •B E t
• 7.
  Theorem: If twosegments are tangent to a circle from the same external point, then the two segments are congruent. • S A N D SD ND≅
• 8.
  Examples The segments ineach figure are tangent. Find the value of each variable. 1. 2. • 2a + 4 5a – 32 • 6y2 18y
• 9.
  Examples The segments ineach figure are tangent. Find the value of each variable. 1. 2. • 2a + 4 5a – 32 • 6y2 18y 5a – 32 = 2a + 4 3a = 36 a = 12 6y2 = 18y y y 6y = 18 y = 3
• 10.
  minorminorminorminor arcarcarcarc –an arc created by a central angle less than 180˚. It can be named with 2 or 3 letters. majormajormajormajor arcarcarcarc – an arc created by a central angle greater than 180˚. Named with 3 letters. semicirclesemicirclesemicirclesemicircle – an arc created by a diameter. (= 180˚) • S P A T (or )SP PS is a minor arc. (or )SAP PAS is a major arc.SPA is a semicircle. • • •
• 11.
  Examples Find the measureof each 1. 2. 3. m∠CAT • • 135˚ F R E D mRE mEFR C A T 260˚
• 12.
  Examples Find the measureof each 1. = 135˚ 2. = 360 – 135 = 225˚ 3. m∠CAT • • 135˚ F R E D mRE mEFR C A T 260˚ = 360 – 260 = 100˚
• 13.
  If a radiusor diameter is perpendicular to a chord, then it bisects the chord and its arc. ER GO⊥ G E O R A GA AO≅ GR RO≅
• 14.
  Example Find the lengthof .BU • B L U E2 5 x
• 15.
  Example Find the lengthof .BU • B L U E 3333 2 5 2 2 2 3 5x+ = x x = 4 BU = 2(4) = 8