Geometry 11.4 Color Theory

11.4 Chords Objectives Determine the relationships between a chord and a diameter of a circle Determine the relationships between congruent chords and their minor arcs Prove the Diameter-Chord Theorem Prove the Equidistant Chord Theorem and the Converse Prove the Congruent-Chord Arc Theorem and the Converse Prove the Segment-Chord Theorem

87 𝑜 180 𝑜 −87 𝑜 = 93 𝑜 93 𝑜 180 𝑜 −93 𝑜 = 87 𝑜

All angles are inscribed (On The Circle) 𝑚 𝑋𝑊 +𝑚 𝑋𝑆 +𝑚 𝑆𝑊 = 360 𝑜 The angle measures are half of the arc measure. 𝑚∠𝑆+𝑚∠𝑊+𝑚∠𝑋= 1 2 360 𝑜 = 180 𝑜 All 3 angles also make a triangle.

Problem 1: Chords and Diameters Discuss #1 and #2 Sketchpad Perpendicular Bisector through Center 1g? The perpendicular bisector of a chord always passes through the center of the circle

Problem 1: Chords and Diameters Diameter-Chord Theorem If a circle’s diameter is perpendicular to a chord, then the diameter bisects the chord and bisects the arc determined by the chord. Reminder: The distance from a point to a segment is the perpendicular segment from the point to the segment.

Problem 1: Chords and Diameters Draw #3 Sketchpad: Equidistant Chords

Problem 1: Chords and Diameters Equidistant Chord Theorem If two chords of the same circle or congruent circles are congruent, then they are equidistant from the center of the circle. Equidistant Chord Converse Theorem If two chords of the same circle or congruent circles are equidistant from the center of the circle, then the chords are congruent.

Problem3: Chords and Arcs Sketchpad: Congruent Chord Congruent Arc Congruent Chord-Congruent Arc Theorem If two chords of the same circle or congruent circles are congruent, then their corresponding arcs are congruent. Congruent Chord-Congruent Arc Converse Theorem If two arcs of the same circle or congruent circles are congruent, then their corresponding chords are congruent.

Problem 4: Segments on Chords Segments of a Chord Segments formed on a chord when two chords of a circle intersect Segments of Chord 𝑌𝑅 𝐸𝑌 𝑎𝑛𝑑 𝐸𝑅 Segments of Chord 𝑃𝑂 𝐸𝑃 𝑎𝑛𝑑 𝐸𝑂

Problem 4: Segments on Chords Segment-Chord Theorem If two chords in a circle intersect, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the second chord. Sketchpad Demo

Collaborate: 2 Minutes 12∙8=10𝑥 96=10𝑥 𝑥=9.6 𝑓𝑒𝑒𝑡

Collaborate: 2 Minutes 𝐵𝐹 || 𝐶𝐸 𝐶𝐻 ≅ 𝐸𝐻 and 𝐶𝐷 ≅ 𝐸𝐷 Diameter-Chord Theorem If a circle’s diameter is perpendicular to a chord, then the diameter bisects the chord and bisects the arc determined by the chord.

Formative Assessment Skills Practice 11.4 Match the Vocabulary Pg. 809 (1-7) Pg. 810-816 (1-24)