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Tangent Theorems

11X1 T13 05 tangent theorems 1 (2011)

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Page 1: 11X1 T13 05 tangent theorems 1 (2011)

Tangent Theorems

Page 2: 11X1 T13 05 tangent theorems 1 (2011)

Tangent Theorems (7) Assumption:

The size of the angle between a tangent and the radius drawn to the

point of contact is 90 degrees.

Page 3: 11X1 T13 05 tangent theorems 1 (2011)

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: 11X1 T13 05 tangent theorems 1 (2011)

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: 11X1 T13 05 tangent theorems 1 (2011)

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: 11X1 T13 05 tangent theorems 1 (2011)

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: 11X1 T13 05 tangent theorems 1 (2011)

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: 11X1 T13 05 tangent theorems 1 (2011)

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: 11X1 T13 05 tangent theorems 1 (2011)

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: 11X1 T13 05 tangent theorems 1 (2011)

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: 11X1 T13 05 tangent theorems 1 (2011)

(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

Page 12: 11X1 T13 05 tangent theorems 1 (2011)

(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

Page 13: 11X1 T13 05 tangent theorems 1 (2011)

(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

Page 14: 11X1 T13 05 tangent theorems 1 (2011)

(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

Page 15: 11X1 T13 05 tangent theorems 1 (2011)

(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

Page 16: 11X1 T13 05 tangent theorems 1 (2011)

(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

Page 17: 11X1 T13 05 tangent theorems 1 (2011)

(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

Page 18: 11X1 T13 05 tangent theorems 1 (2011)

(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: 11X1 T13 05 tangent theorems 1 (2011)

(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

Page 20: 11X1 T13 05 tangent theorems 1 (2011)

(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

Page 21: 11X1 T13 05 tangent theorems 1 (2011)

H

O T

Page 22: 11X1 T13 05 tangent theorems 1 (2011)

collineargent common tan of

contact ofpoint and centres collinear is OTH

H

O T

Page 23: 11X1 T13 05 tangent theorems 1 (2011)

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