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.
OX XY radius tangent
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.
OX XY radius tangent
Z Let x be the distance of the chord from
the centre.
O x X
Y
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.
OX XY radius tangent
Z Let x be the distance of the chord from
the centre.
y Let 2y be the length of the chord.
O x X
y
Y
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.
OX XY radius tangent
Z Let x be the distance of the chord from
the centre.
y Let 2y be the length of the chord.
O x X
line joining centre to
y OX YZ
midpoint, to chord
Y
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.
OX XY radius tangent
Z Let x be the distance of the chord from
the centre.
y Let 2y be the length of the chord.
O x X
line joining centre to
y OX 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 the
point of contact is 90 degrees.
OX XY radius tangent
Let x be the distance of the chord from
Z the centre.
y Let 2y be the length of the chord.
O x X
line joining centre to
y OX YZ
midpoint, to chord
Y
As x radius
y0
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.
OX XY radius tangent
Let x be the distance of the chord from
Z the centre.
y Let 2y be the length of the chord.
O xy X
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 the
point of contact is 90 degrees.
OX XY radius tangent
Let x be the distance of the chord from
Z the centre.
Let 2y be the length of the chord.
O x 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 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
T
O
B
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
AT BT tangents from external point are
A
T
O
B
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
AT BT tangents from external point are
A Prove : AT BT
T
O
B
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
AT BT tangents from external point are
A Prove : AT BT
Proof: Join OA, OB and OT
T
O
B
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
AT BT tangents from external point are
A Prove : AT BT
Proof: Join OA, OB and OT
O T OAT OBT 90 radius tangent
B
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
AT BT tangents from external point are
A Prove : AT BT
Proof: Join OA, OB and OT
O T OAT OBT 90 radius tangent
OT is a common side
B
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
AT BT tangents from external point are
A Prove : AT BT
Proof: Join OA, OB and OT
O T OAT 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 a
circle. The line joining this point to the centre is an axis of
symmetry
AT BT tangents from external point are
A Prove : AT BT
Proof: Join OA, OB and OT
O T OAT OBT 90 radius tangent
OT is a common side
OA OB radii
B
OAT OBT RHS
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
AT BT tangents from external point are
A Prove : AT BT
Proof: Join OA, OB and OT
O T OAT OBT 90 radius tangent
OT is a common side
OA OB radii
B
OAT OBT RHS
AT BT matching sides in 's
22. O T
H
centres and point of contact
OTH is collinear
of common tangent collinear
23. O T
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