GeometryGeometry Chord Lengths Section 6.3 Geometry Mrs. Spitz Spring 2005 Modified By Mr. Moss, Spring 2011

Geometry Geometry Today’s Standards MM2G3. Students will understand the properties of circles. a. Understand and use properties of chords, tangents, and secants as an application of triangle similarity. d. Justify measurements and relationships in circles using geometric and algebraic properties.

Geometry Geometry Using Chords of Circles A point Y is called the midpoint of if . Any line, segment, or ray that contains Y bisects.

Geometry Geometry Theorem 6.5 – covered already 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 Ex. 4: Using Theorem 6.5 You can use Theorem 6.5 to find m Because AD  DC, and . So, m = m 2x = x + 40Substitute x = 40 Subtract x from each side. 2x ° (x + 40) ° D = 2(40) = 80°Substitute

Geometry Geometry Theorem 6.6 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 Ex. 5: Finding the Center of a Circle Theorem 6.6 can be used to locate a circle’s center as shown in the next few slides. Step 1: Draw any two chords that are not parallel to each other.

Geometry Geometry Ex. 5: Finding the Center of a Circle Step 2: Draw the perpendicular bisector of each chord. These are the diameters.

Geometry Geometry Ex. 5: Finding the Center of a Circle Step 3: The perpendicular bisectors intersect at the circle’s center.

Geometry Geometry Theorem 6.7 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. , 

Geometry Geometry Theorem 6.7 Proof If diameter is  to a Chord Then BE  DE Construct radii PB & PD Diameter AC  BD - given PE = PE – reflexive prop PD = PB – both radii of same   PBE   PDE by HL BE  DE by CPCTC

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  AC if and only if DE  GF. Theorem 6.8

Geometry Geometry Draw radii AE and AF DE  FG – given B and C are midpoints BE  CF  EBA =  FCA = 90° AE  AF – both radii  ABE   ACF by HL AB = AC - CPCTC Theorem 6.8 Proof (forwards): If 2 chords are  Then AB = AC

Geometry Geometry Draw radii AE and AF AB = AC – given  EBA =  FCA = 90° AE  AF – both radii  ABE   ACF by HL BE = CF – CPCTC BD = CG by similar DE = FG by addition Theorem 6.8 Proof (reverse): If AB = AC Then 2 chords are 

Geometry Geometry Practice 11-3 Study Guide 11-4 Practice – skip problems 1 & 2 HW: Pg 201, # 1 – 13 all

Geometry Geometry Ex. 7: Using Theorem 6.8 AB = 8; DE = 8, and CD = 5. Find CF.