11 x1 t13 06 tangent theorems 2

17
Alternate Segment Theorem

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Page 1: 11 x1 t13 06 tangent theorems 2

Alternate Segment

Theorem

Page 2: 11 x1 t13 06 tangent theorems 2

Alternate Segment

Theorem (9) An angle formed by a tangent to a circle with a chord drawn to the

point of contact is equal to any angle in the alternate segment.

Page 3: 11 x1 t13 06 tangent theorems 2

Alternate Segment

Theorem (9) An angle formed by a tangent to a circle with a chord drawn to the

point of contact is equal to any angle in the alternate segment.

O

X P

A B

Page 4: 11 x1 t13 06 tangent theorems 2

Alternate Segment

Theorem (9) An angle formed by a tangent to a circle with a chord drawn to the

point of contact is equal to any angle in the alternate segment.

alternate segment theoremBPX PAB

O

X P

A B

Page 5: 11 x1 t13 06 tangent theorems 2

Alternate Segment

Theorem (9) An angle formed by a tangent to a circle with a chord drawn to the

point of contact is equal to any angle in the alternate segment.

alternate segment theoremBPX PAB

O

X P

A B

ABPAPY :Prove

Y

Page 6: 11 x1 t13 06 tangent theorems 2

Alternate Segment

Theorem (9) An angle formed by a tangent to a circle with a chord drawn to the

point of contact is equal to any angle in the alternate segment.

alternate segment theoremBPX PAB

O

X P

A B

ABPAPY :Prove

Proof: Join PO and produce to meet the

circumference at Z

Z

Y

Page 7: 11 x1 t13 06 tangent theorems 2

Alternate Segment

Theorem (9) An angle formed by a tangent to a circle with a chord drawn to the

point of contact is equal to any angle in the alternate segment.

alternate segment theoremBPX PAB

O

X P

A B

ABPAPY :Prove

Proof: Join PO and produce to meet the

circumference at Z Join AZ

Z

Y

Page 8: 11 x1 t13 06 tangent theorems 2

Alternate Segment

Theorem (9) An angle formed by a tangent to a circle with a chord drawn to the

point of contact is equal to any angle in the alternate segment.

alternate segment theoremBPX PAB

O

X P

A B

ABPAPY :Prove

Proof: Join PO and produce to meet the

circumference at Z Join AZ

Z

tangentradius 90 ZPY

Y

Page 9: 11 x1 t13 06 tangent theorems 2

Alternate Segment

Theorem (9) An angle formed by a tangent to a circle with a chord drawn to the

point of contact is equal to any angle in the alternate segment.

alternate segment theoremBPX PAB

O

X P

A B

ABPAPY :Prove

Proof: Join PO and produce to meet the

circumference at Z Join AZ

Z

tangentradius 90 ZPY

Y -90 APYAPZ

Page 10: 11 x1 t13 06 tangent theorems 2

Alternate Segment

Theorem (9) An angle formed by a tangent to a circle with a chord drawn to the

point of contact is equal to any angle in the alternate segment.

alternate segment theoremBPX PAB

O

X P

A B

ABPAPY :Prove

Proof: Join PO and produce to meet the

circumference at Z Join AZ

Z

tangentradius 90 ZPY

Y -90 APYAPZ

semicirclein 90 PAZ

Page 11: 11 x1 t13 06 tangent theorems 2

O

X P

A B

Z

Y

Page 12: 11 x1 t13 06 tangent theorems 2

O

X P

A B

Z

Y

sum 180 APZAPZAZPPAZ

Page 13: 11 x1 t13 06 tangent theorems 2

O

X P

A B

Z

Y

sum 180 APZAPZAZPPAZ

1809090 APYAZP

Page 14: 11 x1 t13 06 tangent theorems 2

O

X P

A B

Z

Y

sum 180 APZAPZAZPPAZ

1809090 APYAZP

APYAZP

Page 15: 11 x1 t13 06 tangent theorems 2

O

X P

A B

Z

Y

sum 180 APZAPZAZPPAZ

1809090 APYAZP

APYAZP

's in same segment =AZP ABP

Page 16: 11 x1 t13 06 tangent theorems 2

O

X P

A B

Z

Y

sum 180 APZAPZAZPPAZ

1809090 APYAZP

APYAZP

's in same segment =AZP ABP

ABPAPY

Page 17: 11 x1 t13 06 tangent theorems 2

O

X P

A B

Z

Y

sum 180 APZAPZAZPPAZ

1809090 APYAZP

APYAZP

's in same segment =AZP ABP

ABPAPY

Exercise 9F; 2ace etc, 3bd, 4bd, 5b, 7b, 9b, 10b, 12, 14, 15b