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GCSE Circle Theorems Dr J Frost Last modified: 19 th February 2014.

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1 GCSE Circle Theorems Dr J Frost Last modified: 19 th February 2014

2 RECAP: Parts of a Circle Sector (Minor) Segment Diameter Radius Tangent Chord (Minor) Arc Circumference ? ? ? ? ?  ? ? ?

3 What are Circle Theorems Circle Theorems are laws that apply to both angles and lengths when circles are involved. We’ll deal with them in groups. #1 Non-Circle Theorems These are not circle theorems, but are useful in questions involving circle theorems ? Angles in a quadrilateral add up to 360. The radius is of constant length Bro Tip: When you have multiple radii, put a mark on each of them to remind yourself they’re the same length.

4 #2 Circle Theorems Involving Right Angles  radius tangent “Angle between radius and tangent is 90  ”. “Angle in semicircle is 90 .” Note that the hypotenuse of the triangle MUST be the diameter. Bro Tip: Remember the wording in the black boxes, because you’re often required to justify in words a particular angle in an exam.

5 #3 Circle Theorems Involving Other Angles  “Angles in same segment are equal.” “Angle at centre is twice the angle at the circumference.” a a a 2a

6 #3 Circle Theorems Involving Other Angles  Opposite angles of cyclic quadrilateral add up to 180. x x 180-x

7 #4 Circle Theorems Involving Lengths Lengths of the tangents from a point to the circle are equal. There’s only one you need to know...

8 Which Circle Theorem? Identify which circle theorems you could use to solve each question. O 160  100  ? Angle in semicircle is 90  Angle between tangent and radius is 90  Opposite angles of cyclic quadrilateral add to 180  Angles in same segment are equal Angle at centre is twice angle at circumference Lengths of the tangents from a point to the circle are equal Two angles in isosceles triangle the same Angles of quadrilateral add to 360  Reveal

9 Which Circle Theorem? Identify which circle theorems you could use to solve each question. 70  60  70  ? Angle in semicircle is 90  Angle between tangent and radius is 90  Opposite angles of cyclic quadrilateral add to 180  Angles in same segment are equal Angle at centre is twice angle at circumference Lengths of the tangents from a point to the circle are equal Two angles in isosceles triangle the same Angles of quadrilateral add to 360  Reveal

10 Which Circle Theorem? Identify which circle theorems you could use to solve each question. 115  ? Angle in semicircle is 90  Angle between tangent and radius is 90  Opposite angles of cyclic quadrilateral add to 180  Angles in same segment are equal Angle at centre is twice angle at circumference Lengths of the tangents from a point to the circle are equal Two angles in isosceles triangle the same Angles of quadrilateral add to 360  Reveal

11 Which Circle Theorem? Identify which circle theorems you could use to solve each question. 70  ? Angle in semicircle is 90  Angle between tangent and radius is 90  Opposite angles of cyclic quadrilateral add to 180  Angles in same segment are equal Angle at centre is twice angle at circumference Lengths of the tangents from a point to the circle are equal Two angles in isosceles triangle the same Angles of quadrilateral add to 360  Reveal

12 Which Circle Theorem? Identify which circle theorems you could use to solve each question. Angle in semicircle is 90  Angle between tangent and radius is 90  Opposite angles of cyclic quadrilateral add to 180  Angles in same segment are equal Angle at centre is twice angle at circumference Lengths of the tangents from a point to the circle are equal 32  ? Two angles in isosceles triangle the same Angles of quadrilateral add to 360  Reveal

13 Which Circle Theorem? Identify which circle theorems you could use to solve each question. Angle in semicircle is 90  Angle between tangent and radius is 90  Opposite angles of cyclic quadrilateral add to 180  Angles in same segment are equal Angle at centre is twice angle at circumference Lengths of the tangents from a point to the circle are equal 31  ? Two angles in isosceles triangle the same Angles of quadrilateral add to 360  Reveal

14 #5 Alternate Segment Theorem This one is probably the hardest to remember and a particular favourite in the Intermediate/Senior Maths Challenges.  The angle between the tangent and a chord...  tangent chord  Click to Start Bromanimation...is equal to the angle in the alternate segment This is called the alternate segment because it’s the segment on the other side of the chord.

15 Check Your Understanding z = 58  ?

16 Check Your Understanding Angle ABC = Give a reason: Angle AOC = Give a reason: Angle CAE = Give a reason: 112  Supplementary angles of cyclic quadrilateral add up to  68  Angle at centre is double angle at circumference. Alternate Segment Theorem. ??? ??? Source: IGCSE Jan 2014 (R)

17 Exercises Printed collection of past GCSE questions.

18 Answers to more difficult questions Determine angle ADB. Source: IGCSE May ? ? ?

19 Answers to more difficult questions (Towards the end of your sheet) ?1?1 ?2?2 ?3?3 Angle at centre is twice angle at circumference Alternate Segment Theorem Two angles in isosceles triangle the same

20 APPENDIX: Proofs A B C O a a 180-2a 2a 90-a Let angle BAO be a. Triangle ABO is isosceles so ABO = a. Remaining angle in triangle must be 180-2a. Thus BOC = 2a. Since triangle BOC is isosceles, angle BOC = OCB = 90 – a. Thus angle ABC = ABO + OBC = a + 90 – a = 90. ? ? ? ? ?

21  x a APPENDIX: Proofs b b a ? ? Opposite angles of cyclic quadrilateral add up to 180. This combined angle = 180 – a – b (angles in a triangle) ? Adding opposite angles: a + b – a – b = 180

22 APPENDIX: Proofs Alternate Segment Theorem 1: Angle between tangent and radius is 90, so angle CAD = 90 -   90-   A B C D ?1?1 ?3?3 ?2?2 2: Angle in semicircle is 90. 3: Angles in triangle add up to 180.  ?4?4 4: But any other angle in the same segment will be the same.


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