Ronald Hui Tak Sun Secondary School HKDSE Mathematics Ronald Hui Tak Sun Secondary School

Missing Homework SHW1-A1 SHW1-B1 Summer Holiday Homework 10 Sep (Last week!) 10, 24 SHW1-B1 14 Sep (Today!) Summer Holiday Homework 25 Sep (Fri) Ronald HUI 14 September 2015

Ronald HUI 14 September 2015

Ronald HUI 14 September 2015

Ronald HUI 14 September 2015

Relationships among Arcs, Chords and Angles

Equal angles at the centre We can summarize the relationships among angles at the centre, arcs and chords of a circle as follows: Equal angles at the centre Equal arcs Equal chords equal s, equal chords equal s, equal arcs equal chords, equal s equal arcs, equal s equal arcs, equal chords equal chords, equal arcs

Arcs Proportional to Angles at the Centre In the figure, OAB is a sector of the circle. If we duplicate sector OAB four times like this: D 40 40 R O A B 40 40 Q P C 40

Arcs Proportional to Angles at the Centre In the figure, OAB is a sector of the circle. If we duplicate sector OAB four times like this: D 160 ∠COD = 160 Obviously, we have: O A B 40 AB : CD = 1 : 4 and ∠AOB : ∠COD = 1 : 4 i.e. AB : CD = ∠AOB : ∠COD C

Theorem 1.16 In a circle, arcs are proportional to their corresponding angles at the centre. O x y D C B A y AB : CD = x : y x Abbreviation: arcs prop. to s at centre

Yes, we can extend Theorem 1.16 as follows. Arcs Proportional to Angles at the Circumference Are arcs proportional to their corresponding angles at the circumference? Yes, we can extend Theorem 1.16 as follows.

By using ‘ at centre twice  at ce’, we can show that Arcs Proportional to Angles at the Circumference By using ‘ at centre twice  at ce’, we can show that AB : CD = m : n. n D C B A m P Q O

Arcs Proportional to Angles at the Circumference Let ∠AOB = x and ∠COD = y. n D C B A m P Q x = 2m  at centre twice  at ce y = 2n  at centre twice  at ce O AB : CD = x : y arcs prop. to s at centre x y = 2m : 2n = m : n

Theorem 1.17 In a circle, arcs are proportional to their corresponding angles at the circumference. n D C B A m P Q AB : CD = m : n n m Abbreviation: arcs prop. to s at ce

arcs prop. to s at centre Example: Find x in the figure. A B C D 80 11 cm 8 cm x O arcs prop. to s at centre ∴

Follow-up question Find x in the figure. ∠CAD = 40° arcs prop. to s at ce Find x in the figure. D 10 8 cm A 40° x ext.  of △ ∠CAD = 40° 2 cm K In △AKD, C B AKC and BKD are straight lines.

Ronald HUI 14 September 2015

Chapter 1 SQ1: 2/10 (Fri) Revision Ex: 30/9 (Wed) Time to work harder please!!! Ronald HUI 14 September 2015