The document discusses several theorems related to tangents of circles: 1) The angle between a tangent line and radius drawn to the point of contact is 90 degrees. 2) From any external point, two equal tangent lines can be drawn to a circle, and the line joining the external point to the center of the circle is an axis of symmetry. 3) The common internal tangent to two circles is the line containing the two centers and the point of contact of the tangent lines.

1. Tangent Theorems

2. Tangent Theorems
(7) Assumption:
The size of the angle between a tangent and the radius drawn to the
point of contact is 90 degrees.

3. Tangent Theorems
(7) Assumption:
The size of the angle between a tangent and the radius drawn to the
point of contact is 90 degrees.
 tangentradius  XYOX

4. Tangent Theorems
(7) Assumption:
The size of the angle between a tangent and the radius drawn to the
point of contact is 90 degrees.
 tangentradius  XYOX
Let x be the distance of the chord from
the centre.
xO X
Y
Z

5. Tangent Theorems
(7) Assumption:
The size of the angle between a tangent and the radius drawn to the
point of contact is 90 degrees.
 tangentradius  XYOX
Let x be the distance of the chord from
the centre.
y
x
Let 2y be the length of the chord.
y
O X
Y
Z

6. Tangent Theorems
(7) Assumption:
The size of the angle between a tangent and the radius drawn to the
point of contact is 90 degrees.
 tangentradius  XYOX
Let x be the distance of the chord from
the centre.
y
x
Let 2y be the length of the chord.
y
O X
Y
Z








chordtomidpoint,
tocentrejoiningline
YZOX

7. Tangent Theorems
(7) Assumption:
The size of the angle between a tangent and the radius drawn to the
point of contact is 90 degrees.
 tangentradius  XYOX
Let x be the distance of the chord from
the centre.
y
x
Let 2y be the length of the chord.
y
O X
Y
Z








chordtomidpoint,
tocentrejoiningline
YZOX
0y
radiusAs

x

8. Tangent Theorems
(7) Assumption:
The size of the angle between a tangent and the radius drawn to the
point of contact is 90 degrees.
 tangentradius  XYOX
Let x be the distance of the chord from
the centre.
y
O x
Let 2y be the length of the chord.
y
X
Y
Z








chordtomidpoint,
tocentrejoiningline
YZOX
0y
radiusAs

x

9. Tangent Theorems
(7) Assumption:
The size of the angle between a tangent and the radius drawn to the
point of contact is 90 degrees.
 tangentradius  XYOX
Let x be the distance of the chord from
the centre.
y
O x
Let 2y be the length of the chord.
y
X
Y
Z








chordtomidpoint,
tocentrejoiningline
YZOX
0y
radiusAs

x

10. Tangent Theorems
(7) Assumption:
The size of the angle between a tangent and the radius drawn to the
point of contact is 90 degrees.
 tangentradius  XYOX
Let x be the distance of the chord from
the centre.
O x
Let 2y be the length of the chord.
X
Y
Z








chordtomidpoint,
tocentrejoiningline
YZOX
0y
radiusAs

x

11. (8) From any external point, two equal tangents may be drawn to a
circle. The line joining this point to the centre is an axis of
symmetry

12. (8) From any external point, two equal tangents may be drawn to a
circle. The line joining this point to the centre is an axis of
symmetry
A
B
O
T

13. (8) From any external point, two equal tangents may be drawn to a
circle. The line joining this point to the centre is an axis of
symmetry
  arepointexternalfromtangentsBTAT
A
B
O
T

14. (8) From any external point, two equal tangents may be drawn to a
circle. The line joining this point to the centre is an axis of
symmetry
  arepointexternalfromtangentsBTAT
A
B
O
T
BTAT :Prove

15. (8) From any external point, two equal tangents may be drawn to a
circle. The line joining this point to the centre is an axis of
symmetry
  arepointexternalfromtangentsBTAT
A
B
O
T
BTAT :Prove
Proof: Join OA, OB and OT

16. (8) From any external point, two equal tangents may be drawn to a
circle. The line joining this point to the centre is an axis of
symmetry
  arepointexternalfromtangentsBTAT
A
B
O
T
BTAT :Prove
Proof: Join OA, OB and OT
 tangentradius90  
OBTOAT

17. (8) From any external point, two equal tangents may be drawn to a
circle. The line joining this point to the centre is an axis of
symmetry
  arepointexternalfromtangentsBTAT
A
B
O
T
BTAT :Prove
Proof: Join OA, OB and OT
 tangentradius90  
OBTOAT
OT is a common side

18. (8) From any external point, two equal tangents may be drawn to a
circle. The line joining this point to the centre is an axis of
symmetry
  arepointexternalfromtangentsBTAT
A
B
O
T
BTAT :Prove
Proof: Join OA, OB and OT
 tangentradius90  
OBTOAT
 radii OBOA
OT is a common side

19. (8) From any external point, two equal tangents may be drawn to a
circle. The line joining this point to the centre is an axis of
symmetry
  arepointexternalfromtangentsBTAT
A
B
O
T
BTAT :Prove
Proof: Join OA, OB and OT
 tangentradius90  
OBTOAT
 radii OBOA
OT is a common side
 RHSOBTOAT 

20. (8) From any external point, two equal tangents may be drawn to a
circle. The line joining this point to the centre is an axis of
symmetry
  arepointexternalfromtangentsBTAT
A
B
O
T
BTAT :Prove
Proof: Join OA, OB and OT
 tangentradius90  
OBTOAT
 radii OBOA
OT is a common side
 RHSOBTOAT 
 matching sides in 'sAT BT   

21. H
O T

22. 





collineargentcommon tanof
contactofpointandcentres
collinearisOTH
H
O T

23. 





collineargentcommon tanof
contactofpointandcentres
collinearisOTH
H
O T
Exercise 9E; 1aceg, 2bdfh, 3ac, 4bd, 6bc, 9, 10ac, 12b, 14, 16a