Chapter 9 Deductive Geometry in Circles

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Deductive Reasoning  The science of deductive reasoning was founded by Aristotle (384 BC BC), an ancient Greek philosopher.  Through deductive reasoning, conclusions drawn from true premises must be true.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles  All apples are fruit. All fruits contain vitamin C. premises  All students in S4A are hardworking. Vincent is a S4A student. premises Deductive Reasoning conclusion  Therefore all apples contain vitamin C. conclusion  Therefore Vincent is a hardworking student.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Every square is a quadrilateral with four equal sides. Every quadrilateral with four equal side is a rhombus. Every square is a rhombus. Deductive Reasoning What is the conclusion?

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Euclid  In deductive geometry, deductive reasoning is used to prove a theorem from axioms or proved theorems.  Euclid (around 365 BC BC), a Greek mathematician, wrote a deductive geometry textbook called “The Elements”.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Starting with axioms and definitions, theorems can be deduced systematically. Deductive Geometry

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles The Converse of a Theorem “If A then B” is a theorem and “If B then A” is proved to be true. “If B then A” is called the converse of the theorem “If A then B”. If ABC is a right-angled triangle with  C  90 , then a 2  b 2  c 2. (Pyth. theorem) In  ABC, if a 2  b 2  c 2, then ABC is a right- angled triangle. (Converse of Pyth. theorem )

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Many theorems about circles can be deduced from theorems about triangles. Deductive Geometry in Circles

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 1

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Proof of Theorem 1 common side given R.H.S.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 2

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles common side given S.S.S. corr.  s,   s adj.  s on st. line Proof of Theorem 2

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 3

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles given radii given R.H.S. corr. sides,   s Proof of Theorem 3

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 4

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles  from centre bisects chord given radii given R.H.S. corr. side,   s Proof of Theorem 4

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 5

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles 2a  2y 2a base  s, isos.  2aext.  of  a  ybase  s, isos.  2a  2yext.  of  Proof of Theorem 5

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 6

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles AOB is a straight line  at centre  2  at  ce 180   2 90  Proof of Theorem 6

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 7

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Proof of Theorem 7 O O Hints:let O be the centre and use theorem 5 Try to prove!

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 8

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles  s at a pt. 180  Proof of Theorem 8 2x  at centre  2  at  ce 2x  2y

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 9

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles x  yproved Proof of Theorem 9

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 10

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles ext.  greater than opp. int.  s given  s in the same segment Proof of Theorem 10

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Proof of Theorem 10 ext.  greater than opp. int.  s given  s in the same segment

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 11

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Proof of Theorem 11 given opp.  s, cyclic quad. ext.  greater than opp. int.  s x  y  180  () x  r  180  ( )  y  r But y  r( )  This is impossible.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles x  y  180  () x  r  180  ( )  y  r But r  y( ) Proof of Theorem 11 given opp.  s, cyclic quad. ext.  greater than opp. int.  s

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 12

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Proof of Theorem 12 The Proof of theorem 12 is similar to the proof of theorem 11. Try to prove!

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 13

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles greater , greater side radii Proof of Theorem 13

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 14

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles radii base  s, isos.  180  sum of  given Proof of Theorem 14 (180    ROT)  2 90 

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 15

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles radii common side 90  tangent  radius R.H.S. corr. sides,  s corr.  s,  s Proof of Theorem 15

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 16

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles  sum of  tangent  radius 90   q  in semi-circle 90   q  s in the same segment Proof of Theorem 16

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles opp.  s, cyclic quad. proved adj.  s on st. line Proof of Theorem 16

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles Theorem 17

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 9 Deductive Geometry in Circles given yext. , cyclic quad. ext.  greater than opp. int.  s Proof of Theorem 17

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