11X1 T13 05 tangent theorems 1 (2011)

Tangent Theorems

Tangent Theorems (7) Assumption:

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

point of contact is 90 degrees.

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

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

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

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

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

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

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

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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

H

O T

collineargent common tan of

contact ofpoint and centres collinear is OTH

H

O T

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