MM2G3 Students will understand properties of circles. MM2G3 d Justify measurements and relationships in circles using geometric and algebraic properties. Apply Properties of Chords Essential Question: How do we use relationships of arcs and chords in a circle? M2 Unit 3: Day 3 Lesson 6.3 Sunday, September 13, 2015

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Daily Homework Quiz Describe each figure as a minor arc, major arc or a semicircle. Find the arc measure. 1. BC ANSWERminor arc, 32 o Daily Homework Quiz

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Daily Homework Quiz 2. Describe each figure as a minor arc, major arc or a semicircle. Find the arc measure. CBE ANSWERmajor arc, 212 o Daily Homework Quiz

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Daily Homework Quiz 3. Describe each figure as a minor arc, major arc or a semicircle. Find the arc measure. BCE ANSWERsemicircle, 180 o Daily Homework Quiz

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 4. Describe each figure as a minor arc, major arc or a semicircle. Find the arc measure. BC AE Explain why = ~. ANSWER BC AE = ~ m AFE = m BFC because the angles are vertical angles, so AFE BFC.Then arcs and are arcs that have the same measure in the same circle. By definition. = ~ AE BC Daily Homework Quiz

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 5. ACD AC Two diameters of P are AB and CD. If m = 50, find m and m.. AD o ANSWER o 310 ; 130 o Daily Homework Quiz

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. radius 1.DC Tell whether the segment is best described as a radius, chord, or diameter of C. Warm Ups diameter 2.BD 3.DE chord 4.AE 5. Solve 4x = 8x – Solve 3x + 2 = 6x – 4. x = 3 x = 2 chord

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Theorem 6.5 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Use congruent chords to find an arc measure In the diagram, P Q, FG JK, and mJK = 80 o. Find mFG SOLUTION Because FG and JK are congruent chords in congruent circles, the corresponding minor arcs FG and JK are congruent. So, mFG = mJK = 80 o.

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. SOLUTION Because AB and BC are congruent chords in the same circle, the corresponding minor arcs AB and BC are congruent. Use the diagram of D. 1. If mAB = 110°, find mBC So, mBC = mAB = 110 o.

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. GUIDED PRACTICE Use the diagram of D. 2. If mAC = 150°, find mAB Because AB and BC are congruent chords in the same circle, the corresponding minor arcs AB and BC are congruent. Subtract Substitute mAB = 105° Simplify So, mBC = mAB And, mBC + mAB + mAC = 360° So, 2 mAB + mAC = 360° 2 mAB + 150° = 360° 2 mAB = 360 – mAB = 210 mAB = 105° ANSWER

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Theorem 6.6 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Theorem 6.7 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Use a diameter SOLUTION Use the diagram of E to find the length of AC. Tell what theorem you use. Diameter BD is perpendicular to AC. So, by Theorem 6.7, BD bisects AC, and CF = AF. Therefore, AC = 2 AF = 2(7) = 14.

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 3. CD So 9x° = (80 – x)° So 10x° = 80° x = 8° So mCD = 9x° = 72° From the diagram Diameter BD is perpendicular to CE. So, by Theorem 6.7, BD bisects CE, Therefore mCD = mDE. Find the measure of the indicated arc in the diagram. SOLUTION

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. 4. DE mCD = mDE. So mDE = 72° 5. CE mCE = mDE + mCD So mCE = 72° + 72° = 144° Find the measure of the indicated arc in the diagram. SOLUTION

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Theorem 6.8 In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. SOLUTION Chords QR and ST are congruent, so by Theorem 6.8 they are equidistant from C. Therefore, CU = CV. CU = CV 2x = 5x – 9 x = 3 So, CU = 2x = 2(3) = 6. Use Theorem 6.8 Substitute. Solve for x. In the diagram of C, QR = ST = 16. Find CU.

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Since CU = CV. Therefore Chords QR and ST are equidistant from center and from theorem 6.8 QR is congruent to ST SOLUTION QR = ST QR = 32 Use Theorem 6.8. Substitute. 6. QR Suppose ST = 32, and CU = CV = 12. Find the given length.

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Since CU is the line drawn from the center of the circle to the chord QR it will bisect the chord. SOLUTION QU = 16 Substitute. 7. QU 2 So QU = QR 1 2 So QU = (32) 1 Suppose ST = 32, and CU = CV = 12. Find the given length.

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Join the points Q and C. Now QUC is right angled triangle. Use the Pythagorean Theorem to find the QC which will represent the radius of the C SOLUTION 8.The radius of C Suppose ST = 32, and CU = CV = 12. Find the given length.

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. SOLUTION Suppose ST = 32, and CU = CV = 12. Find the given length. 8.The radius of C So QC 2 = So QC 2 = So QC 2 = 400 So QC = 20 So QC 2 = QU 2 + CU 2 By Pythagoras Thm Substitute Square Add Simplify ANSWERThe radius of C = 20

MM2G3 Students will understand properties of circles. MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity. Homework Page # 4 – 21 all.