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X2 T08 03 circle geometry (2010)

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X2 T08 03 circle geometry (2010)

  1. 1. Harder Extension 1 Circle Geometry
  2. 2. Harder Extension 1 Circle Geometry Converse Circle Theorems
  3. 3. Harder Extension 1 Circle Geometry Converse Circle Theorems (1) The circle whose diameter is the hypotenuse of a right angled triangle passes through the third vertex.
  4. 4. Harder Extension 1 Circle Geometry Converse Circle Theorems (1) The circle whose diameter is the hypotenuse of a right angled triangle passes through the third vertex. C A B
  5. 5. Harder Extension 1 Circle Geometry Converse Circle Theorems (1) The circle whose diameter is the hypotenuse of a right angled triangle passes through the third vertex. C ABC are concyclic with AB diameter A B
  6. 6. Harder Extension 1 Circle Geometry Converse Circle Theorems (1) The circle whose diameter is the hypotenuse of a right angled triangle passes through the third vertex. C ABC are concyclic with AB diameter  in a semicircle  90   A B
  7. 7. Harder Extension 1 Circle Geometry Converse Circle Theorems (1) The circle whose diameter is the hypotenuse of a right angled triangle passes through the third vertex. C ABC are concyclic with AB diameter  in a semicircle  90   A B (2) If an interval AB subtends the same angle at two points P and Q on the same side of AB, then A,B,P,Q are concyclic.
  8. 8. Harder Extension 1 Circle Geometry Converse Circle Theorems (1) The circle whose diameter is the hypotenuse of a right angled triangle passes through the third vertex. C ABC are concyclic with AB diameter  in a semicircle  90   A B (2) If an interval AB subtends the same angle at two points P and Q on the same side of AB, then A,B,P,Q are concyclic. P Q   A B
  9. 9. Harder Extension 1 Circle Geometry Converse Circle Theorems (1) The circle whose diameter is the hypotenuse of a right angled triangle passes through the third vertex. C ABC are concyclic with AB diameter  in a semicircle  90   A B (2) If an interval AB subtends the same angle at two points P and Q on the same side of AB, then A,B,P,Q are concyclic. P Q   ABQP is a cyclic quadrilateral A B
  10. 10. Harder Extension 1 Circle Geometry Converse Circle Theorems (1) The circle whose diameter is the hypotenuse of a right angled triangle passes through the third vertex. C ABC are concyclic with AB diameter  in a semicircle  90   A B (2) If an interval AB subtends the same angle at two points P and Q on the same side of AB, then A,B,P,Q are concyclic. P Q   ABQP is a cyclic quadrilateral s in same segment are   A B
  11. 11. (3) If a pair of opposite angles in a quadrilateral are supplementary (or if an exterior angle equals the opposite interior angle) then the quadrilateral is cyclic.
  12. 12. (3) If a pair of opposite angles in a quadrilateral are supplementary (or if an exterior angle equals the opposite interior angle) then the quadrilateral is cyclic. The Four Centres Of A Triangle
  13. 13. (3) If a pair of opposite angles in a quadrilateral are supplementary (or if an exterior angle equals the opposite interior angle) then the quadrilateral is cyclic. The Four Centres Of A Triangle (1) The angle bisectors of the vertices are concurrent at the incentre which is the centre of the incircle, tangent to all three sides.
  14. 14. (3) If a pair of opposite angles in a quadrilateral are supplementary (or if an exterior angle equals the opposite interior angle) then the quadrilateral is cyclic. The Four Centres Of A Triangle (1) The angle bisectors of the vertices are concurrent at the incentre which is the centre of the incircle, tangent to all three sides.
  15. 15. (3) If a pair of opposite angles in a quadrilateral are supplementary (or if an exterior angle equals the opposite interior angle) then the quadrilateral is cyclic. The Four Centres Of A Triangle (1) The angle bisectors of the vertices are concurrent at the incentre which is the centre of the incircle, tangent to all three sides.     incentre 
  16. 16. (3) If a pair of opposite angles in a quadrilateral are supplementary (or if an exterior angle equals the opposite interior angle) then the quadrilateral is cyclic. The Four Centres Of A Triangle (1) The angle bisectors of the vertices are concurrent at the incentre which is the centre of the incircle, tangent to all three sides.     incentre incircle 
  17. 17. (2) The perpendicular bisectors of the sides are concurrent at the circumcentre which is the centre of the circumcircle, passing through all three vertices.
  18. 18. (2) The perpendicular bisectors of the sides are concurrent at the circumcentre which is the centre of the circumcircle, passing through all three vertices.
  19. 19. (2) The perpendicular bisectors of the sides are concurrent at the circumcentre which is the centre of the circumcircle, passing through all three vertices. circumcentre
  20. 20. (2) The perpendicular bisectors of the sides are concurrent at the circumcentre which is the centre of the circumcircle, passing through all three vertices. circumcentre circumcircle
  21. 21. (2) The perpendicular bisectors of the sides are concurrent at the circumcentre which is the centre of the circumcircle, passing through all three vertices. circumcentre circumcircle (3) The medians are concurrent at the centroid, and the centroid trisects each median.
  22. 22. (2) The perpendicular bisectors of the sides are concurrent at the circumcentre which is the centre of the circumcircle, passing through all three vertices. circumcentre circumcircle (3) The medians are concurrent at the centroid, and the centroid trisects each median.
  23. 23. (2) The perpendicular bisectors of the sides are concurrent at the circumcentre which is the centre of the circumcircle, passing through all three vertices. circumcentre circumcircle (3) The medians are concurrent at the centroid, and the centroid trisects each median. centroid
  24. 24. (4) The altitudes are concurrent at the orthocentre.
  25. 25. (4) The altitudes are concurrent at the orthocentre.
  26. 26. (4) The altitudes are concurrent at the orthocentre. orthocentre
  27. 27. (4) The altitudes are concurrent at the orthocentre. orthocentre Interaction Between Geometry & Trigonometry
  28. 28. (4) The altitudes are concurrent at the orthocentre. orthocentre Interaction Between Geometry & Trigonometry a b c    diameter if circumcircle sin A sin B sin C
  29. 29. (4) The altitudes are concurrent at the orthocentre. orthocentre Interaction Between Geometry & Trigonometry a b c    diameter if circumcircle sin A sin B sin C Proof: A B C
  30. 30. (4) The altitudes are concurrent at the orthocentre. orthocentre Interaction Between Geometry & Trigonometry a b c    diameter if circumcircle sin A sin B sin C Proof: A P O B C
  31. 31. (4) The altitudes are concurrent at the orthocentre. orthocentre Interaction Between Geometry & Trigonometry a b c    diameter if circumcircle sin A sin B sin C Proof: A A  P  in same segment  P O B C
  32. 32. (4) The altitudes are concurrent at the orthocentre. orthocentre Interaction Between Geometry & Trigonometry a b c    diameter if circumcircle sin A sin B sin C Proof: A A  P  in same segment  P sin A  sin P O B C
  33. 33. (4) The altitudes are concurrent at the orthocentre. orthocentre Interaction Between Geometry & Trigonometry a b c    diameter if circumcircle sin A sin B sin C Proof: A A  P  in same segment  P sin A  sin P PBC  90 O B C
  34. 34. (4) The altitudes are concurrent at the orthocentre. orthocentre Interaction Between Geometry & Trigonometry a b c    diameter if circumcircle sin A sin B sin C Proof: A A  P  in same segment  P sin A  sin P PBC  90  in semicircle  90   O B C
  35. 35. (4) The altitudes are concurrent at the orthocentre. orthocentre Interaction Between Geometry & Trigonometry a b c    diameter if circumcircle sin A sin B sin C Proof: A A  P  in same segment  P sin A  sin P PBC  90  in semicircle  90   BC O B  sin P PC C
  36. 36. (4) The altitudes are concurrent at the orthocentre. orthocentre Interaction Between Geometry & Trigonometry a b c    diameter if circumcircle sin A sin B sin C Proof: A A  P  in same segment  P sin A  sin P PBC  90  in semicircle  90   BC O B  sin P PC BC  PC C sin P
  37. 37. (4) The altitudes are concurrent at the orthocentre. orthocentre Interaction Between Geometry & Trigonometry a b c    diameter if circumcircle sin A sin B sin C Proof: A A  P  in same segment  P sin A  sin P PBC  90  in semicircle  90   BC O B  sin P PC BC BC  PC  PC  sin P sin A C
  38. 38. e.g. (1990) In the diagram, AB is a fixed chord of a circle, P a variable point in the circle and AC and BD are perpendicular to BP and AP respectively.
  39. 39. e.g. (1990) In the diagram, AB is a fixed chord of a circle, P a variable point in the circle and AC and BD are perpendicular to BP and AP respectively. (i) Show that ABCD is a cyclic quadrilateral on a circle with AB as diameter.
  40. 40. e.g. (1990) In the diagram, AB is a fixed chord of a circle, P a variable point in the circle and AC and BD are perpendicular to BP and AP respectively. (i) Show that ABCD is a cyclic quadrilateral on a circle with AB as diameter. BDA  ACB  90
  41. 41. e.g. (1990) In the diagram, AB is a fixed chord of a circle, P a variable point in the circle and AC and BD are perpendicular to BP and AP respectively. (i) Show that ABCD is a cyclic quadrilateral on a circle with AB as diameter. BDA  ACB  90 given 
  42. 42. e.g. (1990) In the diagram, AB is a fixed chord of a circle, P a variable point in the circle and AC and BD are perpendicular to BP and AP respectively. (i) Show that ABCD is a cyclic quadrilateral on a circle with AB as diameter. BDA  ACB  90 given   ABCD is a cyclic quadrilateral
  43. 43. e.g. (1990) In the diagram, AB is a fixed chord of a circle, P a variable point in the circle and AC and BD are perpendicular to BP and AP respectively. (i) Show that ABCD is a cyclic quadrilateral on a circle with AB as diameter. BDA  ACB  90 given   ABCD is a cyclic quadrilateral s in same segment are 
  44. 44. e.g. (1990) In the diagram, AB is a fixed chord of a circle, P a variable point in the circle and AC and BD are perpendicular to BP and AP respectively. (i) Show that ABCD is a cyclic quadrilateral on a circle with AB as diameter. BDA  ACB  90 given   ABCD is a cyclic quadrilateral s in same segment are  AB is diameter as  in semicircle  90
  45. 45. (ii) Show that triangles PCD and APB are similar
  46. 46. (ii) Show that triangles PCD and APB are similar APB  DPC
  47. 47. (ii) Show that triangles PCD and APB are similar APB  DPC common s 
  48. 48. (ii) Show that triangles PCD and APB are similar APB  DPC common s  PDC  PBA
  49. 49. (ii) Show that triangles PCD and APB are similar APB  DPC common s  PDC  PBA exterior cyclic quadrilateral
  50. 50. (ii) Show that triangles PCD and APB are similar APB  DPC common s  PDC  PBA exterior cyclic quadrilateral  PDC ||| PBA
  51. 51. (ii) Show that triangles PCD and APB are similar APB  DPC common s  PDC  PBA exterior cyclic quadrilateral  PDC ||| PBA equiangular 
  52. 52. (iii) Show that as P varies, the segment CD has constant length.
  53. 53. P P A B C D (iii) Show that as P varies, the segment CD has constant length.
  54. 54. P P A B C D (iii) Show that as P varies, the segment CD has constant length. CD PC  AB AP
  55. 55. P P A B C D (iii) Show that as P varies, the segment CD has constant length. CD PC  ratio of sides in ||| s  AB AP
  56. 56. P P A B C D (iii) Show that as P varies, the segment CD has constant length. CD PC  ratio of sides in ||| s  AB AP PC In PCA,  cos P AP
  57. 57. P P A B C D (iii) Show that as P varies, the segment CD has constant length. CD PC  ratio of sides in ||| s  AB AP PC In PCA,  cos P AP CD   cos P AB
  58. 58. P P A B C D (iii) Show that as P varies, the segment CD has constant length. CD PC  ratio of sides in ||| s  AB AP PC In PCA,  cos P AP CD   cos P AB CD  AB cos P
  59. 59. P P A B C D (iii) Show that as P varies, the segment CD has constant length. CD PC  ratio of sides in ||| s  AB AP PC In PCA,  cos P AP CD   cos P AB CD  AB cos P Now, P is constant
  60. 60. P P A B C D (iii) Show that as P varies, the segment CD has constant length. CD PC  ratio of sides in ||| s  AB AP PC In PCA,  cos P AP CD   cos P AB CD  AB cos P Now, P is constant s in same segment are 
  61. 61. P P A B C D (iii) Show that as P varies, the segment CD has constant length. CD PC  ratio of sides in ||| s  AB AP PC In PCA,  cos P AP CD   cos P AB CD  AB cos P Now, P is constant s in same segment are  and AB is fixed
  62. 62. P P A B C D (iii) Show that as P varies, the segment CD has constant length. CD PC  ratio of sides in ||| s  AB AP PC In PCA,  cos P AP CD   cos P AB CD  AB cos P Now, P is constant s in same segment are  and AB is fixed given 
  63. 63. P P A B C D (iii) Show that as P varies, the segment CD has constant length. CD PC  ratio of sides in ||| s  AB AP PC In PCA,  cos P AP CD   cos P AB CD  AB cos P Now, P is constant s in same segment are  and AB is fixed given  CD is constant
  64. 64. (iv) Find the locus of the midpoint of CD.
  65. 65. (iv) Find the locus of the midpoint of CD. ABCD is a cyclic quadrilateral with AB diameter. D C A B
  66. 66. (iv) Find the locus of the midpoint of CD. ABCD is a cyclic quadrilateral with AB diameter. M D C Let M be the midpoint of CD A B
  67. 67. (iv) Find the locus of the midpoint of CD. ABCD is a cyclic quadrilateral with AB diameter. M D C Let M be the midpoint of CD O is the midpoint of AB A B O
  68. 68. (iv) Find the locus of the midpoint of CD. ABCD is a cyclic quadrilateral with AB diameter. M D C Let M be the midpoint of CD O is the midpoint of AB OM is constant A B O
  69. 69. (iv) Find the locus of the midpoint of CD. ABCD is a cyclic quadrilateral with AB diameter. M D C Let M be the midpoint of CD O is the midpoint of AB OM is constant A O B  chords are equidistant from the centre
  70. 70. (iv) Find the locus of the midpoint of CD. ABCD is a cyclic quadrilateral with AB diameter. M D C Let M be the midpoint of CD O is the midpoint of AB OM is constant A O B  chords are equidistant from the centre  M is a fixed distance from O
  71. 71. (iv) Find the locus of the midpoint of CD. ABCD is a cyclic quadrilateral with AB diameter. M D C Let M be the midpoint of CD O is the midpoint of AB OM is constant A O B  chords are equidistant from the centre  M is a fixed distance from O OM 2  OC 2  MC 2
  72. 72. (iv) Find the locus of the midpoint of CD. ABCD is a cyclic quadrilateral with AB diameter. M D C Let M be the midpoint of CD O is the midpoint of AB OM is constant A O B  chords are equidistant from the centre  M is a fixed distance from O OM 2  OC 2  MC 2 2 2 1  1    AB    AB cos P  2  2  1 1  AB  AB 2 cos 2 P 2 4 4 1  AB 2 sin 2 P 4
  73. 73. (iv) Find the locus of the midpoint of CD. ABCD is a cyclic quadrilateral with AB diameter. M D C Let M be the midpoint of CD O is the midpoint of AB OM is constant A O B  chords are equidistant from the centre  M is a fixed distance from O OM 2  OC 2  MC 2 2 2 1  1    AB    AB cos P  2  2  1 1  AB  AB 2 cos 2 P 2 4 4 1  AB 2 sin 2 P 4 1 OM  AB sin P 2
  74. 74. (iv) Find the locus of the midpoint of CD. ABCD is a cyclic quadrilateral with AB diameter. M D C Let M be the midpoint of CD O is the midpoint of AB OM is constant A O B  chords are equidistant from the centre  M is a fixed distance from O  locus is circle, centre O OM 2  OC 2  MC 2 1 2 2 and radius  AB sin P 1  1    AB    AB cos P  2 2  2  1 1  AB  AB 2 cos 2 P 2 4 4 1  AB 2 sin 2 P 4 1 OM  AB sin P 2
  75. 75. 2008 Extension 2 Question 7b) In the diagram, the points P, Q and R lie on a circle. The tangent at P and the secant QR intersect at T. The bisector of PQR meets QR at S so that QPS  RPS   . The intervals RS, SQ and PT have lengths a, b and c respectively.
  76. 76. 2008 Extension 2 Question 7b) In the diagram, the points P, Q and R lie on a circle. The tangent at P and the secant QR intersect at T. The bisector of PQR meets QR at S so that QPS  RPS   . The intervals RS, SQ and PT have lengths a, b and c respectively. (i) Show that TSP  TPS
  77. 77. 2008 Extension 2 Question 7b) In the diagram, the points P, Q and R lie on a circle. The tangent at P and the secant QR intersect at T. The bisector of PQR meets QR at S so that QPS  RPS   . The intervals RS, SQ and PT have lengths a, b and c respectively. (i) Show that TSP  TPS RQP  RPT
  78. 78. 2008 Extension 2 Question 7b) In the diagram, the points P, Q and R lie on a circle. The tangent at P and the secant QR intersect at T. The bisector of PQR meets QR at S so that QPS  RPS   . The intervals RS, SQ and PT have lengths a, b and c respectively. (i) Show that TSP  TPS RQP  RPT  alternate segment theorem 
  79. 79. 2008 Extension 2 Question 7b) In the diagram, the points P, Q and R lie on a circle. The tangent at P and the secant QR intersect at T. The bisector of PQR meets QR at S so that QPS  RPS   . The intervals RS, SQ and PT have lengths a, b and c respectively. (i) Show that TSP  TPS RQP  RPT  alternate segment theorem  TSP  RQP  SPQ
  80. 80. 2008 Extension 2 Question 7b) In the diagram, the points P, Q and R lie on a circle. The tangent at P and the secant QR intersect at T. The bisector of PQR meets QR at S so that QPS  RPS   . The intervals RS, SQ and PT have lengths a, b and c respectively. (i) Show that TSP  TPS RQP  RPT  alternate segment theorem  TSP  RQP  SPQ  exterior ,SPQ 
  81. 81. 2008 Extension 2 Question 7b) In the diagram, the points P, Q and R lie on a circle. The tangent at P and the secant QR intersect at T. The bisector of PQR meets QR at S so that QPS  RPS   . The intervals RS, SQ and PT have lengths a, b and c respectively. (i) Show that TSP  TPS RQP  RPT  alternate segment theorem  TSP  RQP  SPQ  exterior ,SPQ  TSP  RPT  
  82. 82. SPT  RPT  
  83. 83. SPT  RPT    common 
  84. 84. SPT  RPT    common  SPT  TSP
  85. 85. SPT  RPT    common  SPT  TSP 1 1 1 (ii ) Hence show that   a b c
  86. 86. SPT  RPT    common  SPT  TSP 1 1 1 (ii ) Hence show that   a b c
  87. 87. SPT  RPT    common  SPT  TSP 1 1 1 (ii ) Hence show that   a b c TPS is isosceles
  88. 88. SPT  RPT    common  SPT  TSP 1 1 1 (ii ) Hence show that   a b c TPS is isosceles SPT  RPT    2 = 's 
  89. 89. SPT  RPT    common  SPT  TSP 1 1 1 (ii ) Hence show that   a b c TPS is isosceles  2 = 's  SPT  RPT  
  90. 90. SPT  RPT    common  SPT  TSP 1 1 1 (ii ) Hence show that   a b c TPS is isosceles  2 = 's   ST  c SPT  RPT  
  91. 91. SPT  RPT    common  SPT  TSP 1 1 1 (ii ) Hence show that   a b c TPS is isosceles  2 = 's   ST  c  = sides in isosceles   SPT  RPT  
  92. 92. SPT  RPT    common  SPT  TSP 1 1 1 (ii ) Hence show that   a b c TPS is isosceles  2 = 's   ST  c  = sides in isosceles   PT 2  QT  RT SPT  RPT  
  93. 93. SPT  RPT    common  SPT  TSP 1 1 1 (ii ) Hence show that   a b c TPS is isosceles  2 = 's   ST  c  = sides in isosceles   PT 2  QT  RT  square of tangents=products of intercepts  SPT  RPT  
  94. 94. SPT  RPT    common  SPT  TSP 1 1 1 (ii ) Hence show that   a b c TPS is isosceles  2 = 's   ST  c  = sides in isosceles   PT 2  QT  RT  square of tangents=products of intercepts  SPT  RPT   c 2   c  b  c  a 
  95. 95. SPT  RPT    common  SPT  TSP 1 1 1 (ii ) Hence show that   a b c TPS is isosceles  2 = 's   ST  c  = sides in isosceles   PT 2  QT  RT  square of tangents=products of intercepts  SPT  RPT   c 2   c  b  c  a  c 2  c 2  ac  bc  ab
  96. 96. SPT  RPT    common  SPT  TSP 1 1 1 (ii ) Hence show that   a b c TPS is isosceles  2 = 's   ST  c  = sides in isosceles   PT 2  QT  RT  square of tangents=products of intercepts  SPT  RPT   c 2   c  b  c  a  c 2  c 2  ac  bc  ab bc  ac  ab
  97. 97. SPT  RPT    common  SPT  TSP 1 1 1 (ii ) Hence show that   a b c TPS is isosceles  2 = 's   ST  c  = sides in isosceles   PT 2  QT  RT  square of tangents=products of intercepts  SPT  RPT   c 2   c  b  c  a  c 2  c 2  ac  bc  ab bc  ac  ab 1 1 1   a b c
  98. 98. SPT  RPT    common  SPT  TSP 1 1 1 (ii ) Hence show that   a b c TPS is isosceles  2 = 's   ST  c  = sides in isosceles   PT 2  QT  RT  square of tangents=products of intercepts  SPT  RPT   c 2   c  b  c  a  c 2  c 2  ac  bc  ab bc  ac  ab Past HSC Papers 1 1 1   a b c Exercise 10C*

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