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Circle Theorems

Circle Theorems. Euclid of Alexandria Circa 325 - 265 BC. The library of Alexandria was the foremost seat of learning in the world and functioned like a university. The library contained 600 000 manuscripts. O. Parts. Circumference. Major Arc. Tangent. radius. Tangent. Minor Arc.

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Circle Theorems

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  1. Circle Theorems Euclid of Alexandria Circa 325 - 265 BC The library of Alexandria was the foremost seat of learning in the world and functioned like a university. The library contained 600 000 manuscripts. O

  2. Parts Circumference Major Arc Tangent radius Tangent Minor Arc Tangent A Reminder about parts of the Circle Major Segment diameter chord Minor Segment Major Sector Minor Sector

  3. Term’gy yo yo xo o o o xo B B A B A A xo yo Introductory Terminology Arc AB subtends angle x at the centre. Arc AB subtends angle y at the circumference. Chord AB also subtends angle x at the centre. Chord AB also subtends angle y at the circumference.

  4. Th1 Theorem 1 Measure the angles at the centre and circumference and make a conjecture. xo xo xo o o o o yo yo yo yo xo xo xo xo xo o o o o yo yo yo yo

  5. Theorem 1 Measure the angles at the centre and circumference and make a conjecture. xo xo xo o o o o 2xo 2xo 2xo xo 2xo Angle x is subtended in the minor segment. xo xo xo xo o o o o 2xo 2xo 2xo 2xo The angle subtended at the centre of a circle (by an arc or chord) is twice the angle subtended at the circumference by the same arc or chord. (angle at centre) Watch for this one later.

  6. Example Questions Find the unknown angles giving reasons for your answers. 1 2 xo o o 35o yo 84o A A B B angle x = angle y = 42o (Angle at the centre). 70o(Angle at the centre)

  7. Example Questions Find the unknown angles giving reasons for your answers. 3 4 yo 62o B o o qo po xo 42o A B A angle x = angle p = angle q = angle y = (180 – 2 x 42) = 96o (Isos triangle/angle sum triangle). 48o (Angle at the centre) 124o(Angle at the centre) (180 – 124)/2 = 280 (Isos triangle/angle sum triangle).

  8. Th2 Theorem 2 The angle in a semi-circle is a right angle. Find the unknown angles below stating a reason. Diameter o a 30o c angle a = d o angle b = e angle c = angle d = 70o b angle e = This is just a special case of Theorem 1 and is referred to as a theorem for convenience. 90oangle in a semi-circle 90o angle in a semi-circle 20oangle sum triangle 90oangle in a semi-circle 60oangle sum triangle

  9. Th3 Angles subtended by an arc or chord in the same segment are equal. Theorem 3 yo xo xo xo yo xo xo

  10. Theorem 3 Angles subtended by an arc or chord in the same segment are equal. Find the unknown angles in each case yo 38o xo 30o 40o yo xo Angle x = 30o Angle x = angle y = 38o Angle y = 40o

  11. Th4 The angle between a tangent and a radius is 90o. (Tan/rad) Theorem 4 o

  12. The angle between a tangent and a radius is 90o. (Tan/rad) Theorem 4 o

  13. 30o o B xo 36o yo zo T A angle x = angle y = angle z = If OT is a radius and AB is a tangent, find the unknown angles, giving reasons for your answers. 180 – (90 + 36) = 54o Tan/rad and angle sum of triangle. 90o angle in a semi-circle 60oangle sum triangle

  14. Th5 Theorem 5 The Alternate Segment Theorem. The angle between a tangent and a chord through the point of contact is equal to the angle subtended by that chord in the alternate segment. Find the missing angles below giving reasons in each case. xo 60o xo yo zo 45o yo yo xo angle x = angle y = angle z = 45o (Alt Seg) 60o (Alt Seg) 75oangle sum triangle

  15. Th6 Theorem 6 Cyclic Quadrilateral Theorem. x p y s q r z w The opposite angles of a cyclic quadrilateral are supplementary. (They sum to 180o) Angles p + q = 180o Angles x + w = 180o Angles y + z = 180o Angles r + s = 180o

  16. Theorem 6 Cyclic Quadrilateral Theorem. The opposite angles of a cyclic quadrilateral are supplementary. (They sum to 180o) 70o Find the missing angles below given reasons in each case. r x y q 110o p 85o 135o angle x = angle p = angle y = angle q = angle r = 180 – 85 = 95o(cyclic quad) 180 – 135 = 45o(straight line) 180 – 70 = 110o(cyclic quad) 180 – 110 = 70o (cyclic quad) 180 – 45 = 135o(cyclic quad)

  17. Th7 Theorem 7 Two Tangent Theorem. R P Q R Q U T T PT = PQ PT = PQ P U From any point outside a circle only two tangents can be drawn and they are equal in length.

  18. Theorem 7 Two Tangent Theorem. From any point outside a circle only two tangents can be drawn and they are equal in length. angle w = angle x = angle y = angle z = PQ and PT are tangents to a circle with centre O. Find the unknown angles giving reasons. yo Q xo O 90o (tan/rad) 98o 90o (tan/rad) 49o (angle at centre) 360o – 278 = 82o(quadrilateral) zo wo T P

  19. Theorem 7 Two Tangent Theorem. From any point outside a circle only two tangents can be drawn and they are equal in length. angle w = angle x = angle y = angle z = PQ and PT are tangents to a circle with centre O. Find the unknown angles giving reasons. zo Q O yo 90o (tan/rad) xo 180 – 140 = 40o(angles sum tri) 50o (isos triangle) 50o(alt seg) 80o wo 50o T P

  20. Th8 Theorem 8 Chord Bisector Theorem. A line drawn perpendicular to a chord and passing through the centre of a circle, bisects the chord.. Find length OS O O 3 cm S T 8 cm OS = 5 cm (pythag triple: 3,4,5)

  21. Theorem 8 Chord Bisector Theorem. A line drawn perpendicular to a chord and passing through the centre of a circle, bisects the chord.. Find angle x O O 22o xo U S T Angle SOT = 22o (symmetry/congruenncy) Angle x = 180 – 112 = 68o (angle sum triangle)

  22. Mixed Q 1 Mixed Questions U PTR is a tangent line to the circle at T. Find angles SUT, SOT, OTS and OST. O S R 65o T Angle SUT = Angle SOT = P Angle OTS = Angle OST = 65o (Alt seg) 130o (angle at centre) 25o (tan rad) 25o (isos triangle)

  23. Mixed Q 2 Mixed Questions PR and PQ are tangents to the circle. Find the missing angles giving reasons. Q U P y 110o O w z x 48o R Angle w = Angle x = Angle y = Angle z = 22o (cyclic quad) 68o (tan rad) 44o (isos triangle) 68o (alt seg)

  24. Worksheet 3 The angle subtended by an arc or chord at the centre of a circle is twice the angle subtended at the circumference by the same arc or chord. A O B C Theorem 1 and 2

  25. Worksheet 4 D O B C Angles subtended by an arc or chord in the same segment are equal. A Theorem 3

  26. Worksheet 5 The angle between a tangent and a radius drawn to the point of contact is a right angle. O B T A Theorem 4

  27. Worksheet 6 The angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment. D B O C T A Theorem 5

  28. The opposite angles of a cyclic quadrilateral are supplementary (Sum to 180o). B A C D Theorem 6

  29. Worksheet 8 The two tangents drawn from a point outside a circle are of equal length. Theorem 7 A P O O A B B Theorem 8 A line, drawn perpendicular to a chord and passing through the centre of a circle, bisects the chord. C

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