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Geometry/Trig 2 Name: __________________________ Unit 8 GSP Explorations & Notes Date: ___________________________. Corollary 1. Conclusion (Corollary 1): Segments that are tangent to a circle from a point are ___________________. Sketch the diagram: Fill in the Measurements:. Example 1:.
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Geometry/Trig 2 Name: __________________________ Unit 8 GSP Explorations & Notes Date: ___________________________ Corollary 1 Conclusion (Corollary 1): Segments that are tangent to a circle from a point are ___________________. Sketch the diagram: Fill in the Measurements: Example 1: B A B and C are points of tangency. What type of triangle is DBAC? _______________________ mÐBAC = 32 mÐABC = ________ mÐBCA = ________ C Example 2: B B and C are points of tangency. ½x + 9 x = __________ BA = _________ CA = _________ A 4x + 2 C Theorem 1 Conclusion (Theorem 1): In the same circle or in congruent circles, congruent chords intercept ___________________ arcs. Sketch the diagram: Fill in the Measurements: Examples: Find all angle and arc measures. C mÐCAB = 40 mÐACB = ________ mÐABC = _______ mAB = 140 mAC = __________ mCB = _________ A B D Q is the center of the circle. mAB = 86 mDC = _______ mÐDQC = ______ Classify DDQC by sides: _____________ mBC = 128 mBAC = ___________ A C Q B Complete Quia Quiz Check for Understanding Circles Unit 1
Geometry/Trig 2 Name: __________________________ Unit 8 GSP Explorations & Notes – page 2 Date: ___________________________ Theorem 2 Conclusion (Theorem 2): A diameter that is perpendicular to a chord _________________ the chord and its intercepted arc. Sketch the diagram: Fill in the Measurements: Examples (Q is the center of each circle). S RT = _________ QM = ________ QS = _________ MS = ________ SP = __________ 15 R T M 17 Q P mAB = ________ mAC = _________ mCB = _______ mÐAQC = ________ mÐAQB = _______ mÐABQ = ______ A D F C Q B Challenge: If QC = 10, find AB. mADB = 220 Theorem 3 To measure the distance between a point and a segment, you must measure the _______________________________ distance. Sketch the diagram: Fill in the Measurements: Conclusion (Theorem 3): In the same circle or in congruent circles, ___________________ chords are equally distant from the center. Example (Q is the center of the circle). Given: QJ = QL = 3; KP = 8 JP = _______ NM = _______ LM = _______ LN = ________ QM = _______ QK = ________ (d) mÐQNL = __________ You will need to draw in QM, QK, and QN to complete this problem. Complete Quia Quiz Check for Understanding Circles Unit 2
Geometry/Trig 2 Name: __________________________ Unit 8 GSP Explorations & Notes – page 3 Date: ___________________________ Theorem 4 Inscribed Angle: _________________________________________________________ ____________________________________________________________________________________________________________________________________________ Sketch the Diagram Fill in the Measurements: Conclusion (Theorem 4): The measure of an inscribed angle is equal to ____________________________________________ of its intercepted arc. Example: mÐGFJ = ________ mHJ = __________ mFG = __________ mFGH = _________ mFHG = _________ G F 92° 44° J 109° H Corollary 2 Conclusion (Corollary 2): Inscribed Angles that intercept the same arc are ___________________________. Sketch the diagram: Fill in the Measurements: Example: mAE = 102 mÐABE = __________ mÐACE = __________ mÐADE = __________ mBD = 129 mÐBAD = __________ Complete Quia Quiz Check for Understanding Circles Unit 3
Geometry/Trig 2 Name: __________________________ Unit 8 GSP Explorations & Notes – page 4 Date: ___________________________ Corollary 3 Sketch the Diagram (include measurement): Conclusion (Corollary 3): An angle inscribed inside of a semicircle is ___________________________________. Examples: (AB is a diameter of each circle). (Round all decimal answers to the nearest tenth.) mBD = 80 mÐADB = _____ mÐACB = _____ w = _________ x = __________ y = _________ z = __________ y° w° z° x° Complete: AB is a _______________. ACB is a _______________. D AB = 26, AD = 24, DB = ________ mÐDBA = ______ mÐDAB = _____ B A Corollary 4 Sketch the Diagram (include four angle measurements): Conclusion (Corollary 4): If a quadrilateral is inscribed in a circle, then its opposite angles are _____________________. Example: Find: mÐJKL = __________ mÐKLM = __________ mMJK = ___________ mJK = _____________ mMLK = ____________ mLMJ = ____________ mLMK = ____________ mÐLMJ = 73 mÐMJK = 88 mMJ = 102 Complete Quia Quiz Check for Understanding Circles Unit 4
Geometry/Trig 2 Name: __________________________ Unit 8 GSP Explorations & Notes – page 5 Date: ___________________________ Theorem 5 Conclusion (Theorem 5): The measure of an angle formed by a chord and a tangent is equal to __________________________ _____________________________ of the intercepted arc. Sketch the Diagram: Fill in the Measurements: Example: mÐDBC = 78 mDB = ____________ mDFB = ___________ mÐABD = __________ D F B C A B is a point of tangency. Theorem 6 RULE: Angle = ½(Bigger Arc – Smaller Arc) Case 1 – Two Secants Case 2 – Two Tangents Case 3 – A Secant & A Tangent 1 2 3 mÐ1 = _________________ mÐ2 = ________________ mÐ3 = ________________ Example 1: mÐCAB = 20 mDB = 115 mCB = _________ mCD = _________ mCDB = ________ mBCD = ________ Example 2: mBC = 116 mBDC = ________ mÐCAB = _______ D B C D A B A C B is a point of tangency. B and C are points of tangency. Complete Quia Quiz Check for Understanding Circles Unit 5
Geometry/Trig 2 Name: __________________________ Unit 8 GSP Explorations & Notes – page 6 Date: ___________________________ Answers to the Example Problems Corollary 1 Theorem 1 mÐABC = 74 mÐBCA = 74 mÐACB = 70 mÐABC = 70 mAC = 140 mCB = 80 Example 1: Example 1: mDC = 86 mÐDQC = 86 Classify DDQC by sides: Isosceles mBAC = 232 x = 2 BA = 10 CA = 10 Example 2: Example 2: Theorem 2 Theorem 3 RT = 30 QM = 8 QS = 17 MS = 9 SP = 34 Example 1: JP = 4 NM = 8 LM = 4 LN = 4 QM = 5 QK = 5 (d) mÐQNL = 36.9 mAB = 140 mAC = 70 mCB = 70 mÐAQC = 70 mÐAQB = 140 mÐABQ = 20 Challenge: AB = 18.8 Example 2: mÐGFJ = 46 mHJ = 88 mFG = 71 mFGH = 251 mFHG = 289 Theorem 4 Corollary 2 mÐABE = 51 mÐACE = 51 mÐADE = 51 mÐBAD = 64.5 Corollary 3 Corollary 4 mÐADB = 90 mÐACB = 90 w = 40 x = 40 y = 50 z = 50 mÐJKL = 107 mÐKLM = 92 mMJK = 184 mJK = 82 Example 1: mMLK = 176 mLMJ = 214 mLMK = 278 Example 2: AB = 26, AD = 24, DB = 10 mÐDBA = 67.4 mÐDAB = 22.6 Theorem 5 Theorem 6 mDB = 156 mDFB = 204 mÐABD = 102 mCB = 75 mCD = 170 mCDB = 285 mBCD = 245 mBDC = 244 mÐCAB = 64
