11 x1 t13 05 tangent theorems 1 (2013)
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Tangent Theorems
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Tangent Theorems (7) Assumption:
The size of the angle between a tangent and the radius drawn to the
point of contact is 90 degrees.
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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
![Page 4: 11 x1 t13 05 tangent theorems 1 (2013)](https://reader034.fdocuments.in/reader034/viewer/2022051515/559470251a28ab990e8b4683/html5/thumbnails/4.jpg)
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.
x O X
Y
Z
![Page 5: 11 x1 t13 05 tangent theorems 1 (2013)](https://reader034.fdocuments.in/reader034/viewer/2022051515/559470251a28ab990e8b4683/html5/thumbnails/5.jpg)
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
![Page 6: 11 x1 t13 05 tangent theorems 1 (2013)](https://reader034.fdocuments.in/reader034/viewer/2022051515/559470251a28ab990e8b4683/html5/thumbnails/6.jpg)
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
chord tomidpoint,
tocentre joining line YZOX
![Page 7: 11 x1 t13 05 tangent theorems 1 (2013)](https://reader034.fdocuments.in/reader034/viewer/2022051515/559470251a28ab990e8b4683/html5/thumbnails/7.jpg)
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
chord tomidpoint,
tocentre joining line YZOX
0y
radius As
x
![Page 8: 11 x1 t13 05 tangent theorems 1 (2013)](https://reader034.fdocuments.in/reader034/viewer/2022051515/559470251a28ab990e8b4683/html5/thumbnails/8.jpg)
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
chord tomidpoint,
tocentre joining line YZOX
0y
radius As
x
![Page 9: 11 x1 t13 05 tangent theorems 1 (2013)](https://reader034.fdocuments.in/reader034/viewer/2022051515/559470251a28ab990e8b4683/html5/thumbnails/9.jpg)
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
chord tomidpoint,
tocentre joining line YZOX
0y
radius As
x
![Page 10: 11 x1 t13 05 tangent theorems 1 (2013)](https://reader034.fdocuments.in/reader034/viewer/2022051515/559470251a28ab990e8b4683/html5/thumbnails/10.jpg)
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
chord tomidpoint,
tocentre joining line YZOX
0y
radius As
x
![Page 11: 11 x1 t13 05 tangent theorems 1 (2013)](https://reader034.fdocuments.in/reader034/viewer/2022051515/559470251a28ab990e8b4683/html5/thumbnails/11.jpg)
(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
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(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
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(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
arepoint external from tangents BTAT
A
B
O T
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(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
arepoint external from tangents BTAT
A
B
O T
BTAT :Prove
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(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
arepoint external from tangents BTAT
A
B
O T
BTAT :Prove
Proof: Join OA, OB and OT
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(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
arepoint external from tangents BTAT
A
B
O T
BTAT :Prove
Proof: Join OA, OB and OT
tangentradius 90 OBTOAT
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(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
arepoint external from tangents BTAT
A
B
O T
BTAT :Prove
Proof: Join OA, OB and OT
tangentradius 90 OBTOAT
OT is a common side
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(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
arepoint external from tangents BTAT
A
B
O T
BTAT :Prove
Proof: Join OA, OB and OT
tangentradius 90 OBTOAT
radii OBOA
OT is a common side
![Page 19: 11 x1 t13 05 tangent theorems 1 (2013)](https://reader034.fdocuments.in/reader034/viewer/2022051515/559470251a28ab990e8b4683/html5/thumbnails/19.jpg)
(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
arepoint external from tangents BTAT
A
B
O T
BTAT :Prove
Proof: Join OA, OB and OT
tangentradius 90 OBTOAT
radii OBOA
OT is a common side
RHSOBTOAT
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(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
arepoint external from tangents BTAT
A
B
O T
BTAT :Prove
Proof: Join OA, OB and OT
tangentradius 90 OBTOAT
radii OBOA
OT is a common side
RHSOBTOAT
matching sides in 'sAT BT
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H
O T
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collineargent common tan of
contact ofpoint and centres collinear is OTH
H
O T
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collineargent common tan of
contact ofpoint and centres collinear is OTH
H
O T
Exercise 9E; 1aceg, 2bdfh, 3ac, 4bd, 6bc, 9, 10ac, 12b, 14, 16a