Sec. 12 – 2 Chords and Arcs Objectives: 1) To use  chords, arcs, & central  s. 2) To recognize properties of lines through the center of a circle.

More Circle Properties C Q P R Chord – A segment whose endpts are on a circle. Ex: PQ Central  s –  in a circle, whose vertex is at the center of the circle. Rays of central  s are radii of the circle. Sum of central  s (w/ no common interior pts) are 360°

Central  s and Arcs A central  will divide a circle into 2 arcs. Minor Arc – –Less than ½ of the circle – –Meas. is always less than (<) 180°. – –Name it using 2 letters – –Ex: PQ Major Arc – –More than ½ of the circle – –Meas. is always more than (>) 180°. – –Name it using 3 letters – –Ex: PRQ C Q P R ** All arcs are measured by their corresponding central  s.

Things to remember about Arcs  Central  s have  chords.  Chords have  Arcs.  Arcs have  Central  s. Semicircle –When the measure of an arc = 180°. –Cuts the circle in half. –Name it using 3 letters.

Ex.1: Naming Arcs Name a minor arc. Name a major arc. Name a semicircle. T A B C AB, BC, CD, or BD ADB, DBA, ABD ADC or CBA D

Ex.2: Meas. of arcs and central  T A B D 22° C mDC = m  ATB = mAB = m  ATD = mAD = mADC = m  ATC 22° 90° 158° 180°

More with Chords 1) Chords equidistant from the center are . - If TP  RP, then AB  CD - If TP  RP, then AB  CD 2)  chords are equidistant from the center. - If CD  AB, then TP  RP 3) If 1 & 2 are true, then TP bisects AB & RP bisects CD. - If AT  BT, then CR  DR - If AT  BT, then CR  DR A T B P R D C ll

Ex.3: Solve for the missing Variables m  B = 32  A B P D l l l l 9cm 12.5cm 16  AB = m  P = m  1 = m  2 = BP = 25cm 148  74  15.4cm a 2 + b 2 = c = BP = BP 2 BP = 15.4cm

3 more special appications In a circle, a diameter that is  to a chord bisects the chord & its arc. In a circle, a diameter that bisects a chord (that is not a diameter) is  to the chord. In a circle, the  bisector of a chord contains the center of the circle.

Ex.4: Solve for the missing sides. A B D C 7m 3m BC = AB = AD = 7m 14m 7.6m a 2 + b 2 = c = AD = AD 2 AD = 7.6m

What have we learned??? Central  measure = arc measure Minor Arc –Naming using 2 letters (<180  ) Major Arc –Naming using 3 letters (>180  ) The  bisector of the circle contains the center of the circle. A T B C D 123 