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