11x1 t13 05 tangent theorems 1 (2011)

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  • 1. Tangent Theorems

2. Tangent Theorems(7) Assumption:The size of the angle between a tangent and the radius drawn to thepoint of contact is 90 degrees. 3. Tangent Theorems(7) Assumption:The size of the angle between a tangent and the radius drawn to thepoint of contact is 90 degrees.OX XYradius tangent 4. Tangent Theorems(7) Assumption:The size of the angle between a tangent and the radius drawn to thepoint of contact is 90 degrees.OX XYradius tangent Z Let x be the distance of the chord fromthe centre.O xXY 5. Tangent Theorems(7) Assumption:The size of the angle between a tangent and the radius drawn to thepoint of contact is 90 degrees.OX XYradius tangent ZLet x be the distance of the chord fromthe centre. yLet 2y be the length of the chord.O xXy Y 6. Tangent Theorems(7) Assumption:The size of the angle between a tangent and the radius drawn to thepoint of contact is 90 degrees.OX XYradius tangent ZLet x be the distance of the chord fromthe centre. yLet 2y be the length of the chord.O xX line joining centre to yOX YZ midpoint, to chord Y 7. Tangent Theorems(7) Assumption:The size of the angle between a tangent and the radius drawn to thepoint of contact is 90 degrees.OX XYradius tangent ZLet x be the distance of the chord fromthe centre. yLet 2y be the length of the chord.O xX line joining centre to yOX YZ midpoint, to chord Y As x radius y0 8. Tangent Theorems(7) Assumption:The size of the angle between a tangent and the radius drawn to thepoint of contact is 90 degrees.OX XYradius tangent Let x be the distance of the chord fromZ the centre.y Let 2y be the length of the chord.OxX line joining centre to y OX YZ midpoint, to chord YAs x radiusy0 9. Tangent Theorems(7) Assumption:The size of the angle between a tangent and the radius drawn to thepoint of contact is 90 degrees.OX XYradius tangent Let x be the distance of the chord from Zthe centre. yLet 2y be the length of the chord.OxyX line joining centre to OX YZ midpoint, to chord Y As x radius y0 10. Tangent Theorems(7) Assumption:The size of the angle between a tangent and the radius drawn to thepoint of contact is 90 degrees.OX XYradius tangent Let x be the distance of the chord from Z the centre. Let 2y be the length of the chord.Ox X line joining centre to OX YZ midpoint, to chord Y As x radius y0 11. (8) From any external point, two equal tangents may be drawn to acircle. The line joining this point to the centre is an axis ofsymmetry 12. (8) From any external point, two equal tangents may be drawn to acircle. The line joining this point to the centre is an axis ofsymmetryAT O B 13. (8) From any external point, two equal tangents may be drawn to acircle. The line joining this point to the centre is an axis ofsymmetry AT BT tangents from external point are AT O B 14. (8) From any external point, two equal tangents may be drawn to acircle. The line joining this point to the centre is an axis ofsymmetry AT BT tangents from external point are AProve : AT BTT O B 15. (8) From any external point, two equal tangents may be drawn to acircle. The line joining this point to the centre is an axis ofsymmetry AT BT tangents from external point are AProve : AT BT Proof: Join OA, OB and OTT O B 16. (8) From any external point, two equal tangents may be drawn to acircle. The line joining this point to the centre is an axis ofsymmetry AT BT tangents from external point are AProve : AT BT Proof: Join OA, OB and OT OTOAT OBT 90 radius tangent B 17. (8) From any external point, two equal tangents may be drawn to acircle. The line joining this point to the centre is an axis ofsymmetry AT BT tangents from external point are AProve : AT BT Proof: Join OA, OB and OT OTOAT OBT 90 radius tangent OT is a common side B 18. (8) From any external point, two equal tangents may be drawn to acircle. The line joining this point to the centre is an axis ofsymmetry AT BT tangents from external point are AProve : AT BT Proof: Join OA, OB and OT OTOAT OBT 90 radius tangent OT is a common side OA OB radii B 19. (8) From any external point, two equal tangents may be drawn to acircle. The line joining this point to the centre is an axis ofsymmetry AT BT tangents from external point are AProve : AT BT Proof: Join OA, OB and OT OTOAT OBT 90 radius tangent OT is a common side OA OB radii B OAT OBTRHS 20. (8) From any external point, two equal tangents may be drawn to acircle. The line joining this point to the centre is an axis ofsymmetry AT BT tangents from external point are AProve : AT BT Proof: Join OA, OB and OT OTOAT OBT 90 radius tangent OT is a common side OA OB radii B OAT OBTRHS AT BT matching sides in s 21. O TH 22. O T H centres and point of contact OTH is collinear of common tangent collinear 23. OT H centres and point of contact OTH is collinear of common tangent collinear Exercise 9E; 1aceg, 2bdfh, 3ac, 4bd, 6bc, 9, 10ac, 12b, 14, 16a