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RONALD HUI TAK SUN SECONDARY SCHOOL

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Presentation on theme: "RONALD HUI TAK SUN SECONDARY SCHOOL"— Presentation transcript:

1 RONALD HUI TAK SUN SECONDARY SCHOOL
HKDSE Mathematics RONALD HUI TAK SUN SECONDARY SCHOOL

2 Missing Homework SHW1-A1 (Re-do) SHW1-D1 SHW1-R1 SHW1-P1 12 9, 12
5J07, 9, 10, 12, 14, 19, 20, 23 SHW1-P1 23 Ronald HUI 2 October 2015

3 Missing Homework RE1 Summer Holiday Homework SQ1 5J07, 23, 24
Last Tuesday!!!! SQ1 Next Tuesday (6 Oct) Ronald HUI 2 October 2015

4 Tangents to a Circle and their Properties

5 If you draw a straight line, how many intersections
do the line and the circle have? O

6 There are 3 possible cases.
Q A B O x Q P T O x Q P O x 2 intersections 1 intersection no intersections PQ intersects the circle at two distinct points A and B. PQ intersects the circle at only one point T. PQ does not intersect the circle.

7 PQ touches the circle at T.
In particular, in case 2, we say that O PQ touches the circle at T. P Q PQ is called the tangent to the circle at T. tangent T point of contact T is called the point of contact. For any point on the circle, we can draw one and only one tangent passing through it.

8 Radii and Tangents If PQ is the tangent to the circle at T,
what is the relationship between the radius OT and the tangent PQ?

9 Consider a line PQ which cuts
Radii and Tangents Consider a line PQ which cuts the circle at A and B. O P Q A B T

10 Radii and Tangents ∠OMB = 90 If PQ intersects OT at M such that
AM = BM, then PQ⊥OT. ∠OMB = 90 O (line joining centre to mid-pt. of chord ⊥ chord) M P Q A B T

11 Radii and Tangents If we move PQ downwards such that the mid-point M always lie on OT, until A, M and B all coincide at T, O M P Q A B T

12 Radii and Tangents ∠OTQ = 90
If we move PQ downwards such that the mid-point M always lie on OT, until A, M and B all coincide at T, then PQ becomes the tangent to the circle at T. O ∠OTQ = 90 P Q T

13 Theorem 2.1 If PQ is the tangent to the circle at T, then PQ  OT. O P
Abbreviation: tangent  radius O P Q T

14 Let us use Theorem 2.1 to solve the following problem.
In the figure, PQ is the tangent to the circle at T. Find x. Q P O T x 40 tangent  radius ∵ ÐOTQ = 90 ∴ x + 40 = 90 x = 50

15 In the figure, OT is a radius of the circle and PQ  OT. O
P Q T Is PQ the tangent to the circle at T? Yes, the converse of Theorem 2.1 is also true.

16 Theorem 2.2 (Converse of Theorem 2.1)
Let OT be a radius of the circle and PTQ be a straight line. If PQ  OT, then PQ is the tangent to the circle at T. Abbreviation: converse of tangent  radius O P Q T

17 Refer to the figure. Is AB the tangent to the circle at P?
Example: Refer to the figure. Is AB the tangent to the circle at P? A B C D O P = 90 alt. s, AB // CD ∴ AB is the tangent to the circle at P. converse of tangent  radius

18 converse of tangent ⊥ radius
Follow-up question In the figure, TP is the tangent to the circle at P. Prove that TQ is the tangent to the circle at Q. tangent  radius ÐOPT = 90 P In quadrilateral OQTP, ÐOQT + 50 + 90 + 130 = 360 O 130 ∠ sum of polygon ÐOQT = 90 50 ∴ TQ is the tangent to the circle at Q. T Q converse of tangent ⊥ radius

19 PQ is the tangent to the circle at T. PQ  OT
In summary, we have: OT is a radius. Theorem 2.1 PQ is the tangent to the circle at T. PQ  OT Theorem 2.2 (converse of Theorem 2.1) O P Q T

20 From Theorems 2.1 and 2.2, the theorem below follows directly.
at its point of contact T The perpendicular to a tangent PQ passes through the centre O of the circle. P Q Abbreviation:  to tangent at its point of contact passes through centre O T

21 2 October 2015 Ronald HUI


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