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### Parallel Lines and Circles

### Inscribed Angles (SS on C)

### Theorem (ST on C)

### Theorem

### Theorem

### The Two-Secant Power Theorem.

### Two Secant Power Theorem: QR · QS = QU · QT

### The Tangent - Secant Power Theorem.

### The Tangent - Secant Power Theorem.

### The Two-Chord Power Theorem.

### The Two-Chord Power Theorem.

### The Two-Chord Power Theorem.

WARM UP EXERCSE

B

Q

20

12

P

R

A

16

C

∆ABC with sides 12, 16, 20 is circumscribed about a circle with points of tangency P, Q, R. Find the radius of the circle.

WARM UP EXERCSE

B

Q

20

12

P

R

A

16

C

∆ABC with sides 12, 16, 20 is circumscribed about a circle with points of tangency P, Q, R. Find the radius of the circle.

Hint: it is a right triangle!

20 - x

x

The lines were are going to consider are tangent lines, and secant lines which contain chords of the circle.

We will begin our study with parallel lines. That is, lines which do not intersect.

Most of the theorems will use information from the last class and not triangles.

Parallel Lines and Circles5

Theorem: Parallel lines intercept equal arcs on a circle.

C

D

C

D

C

D

B

A

A

B

A

B

There are three cases: a tangent and a secant, two secants, and two tangents.

6

Tangent-Secant Proof

O

P

D

C

A

B

E

Given: AB ‖ CD and AB tangent at E.

Prove: arc CE = arc DE

Think!

The other cases can be reduced to this case.

7

We will now move on to non-parallel lines. These line (tangents & secants) may intersect on the circle, or inside the circle or outside the circle.

Non Parallel Lines and CirclesLet’s begin with the case where the lines intersect on the circle.

8

Angle ABC is an inscribed angle of a circle if AB and BC are chords of the circle.

A

A

A

O

O

O

B

B

B

C

C

C

Theorem: If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. There are three cases:

9

Proof

A

Given: Inscribed angle ABC.

Prove: ABC= ½ arc AC

Case 1: O on angle.

O

B

C

Note: This theorem implies that an angle inscribed in a semicircle is a right angle.

10

An angle formed by a chord and a tangent at one end of the chord is half the intercepted arc.

D

C

B

A

11

An angle formed by two intersecting chords is half the sum of the two intercepted arcs.

C

E

B

A

D

15

An angle formed by two secants, by a secant and tangent, or by two tangents is half the difference of the intercepted arcs.

19

We will now move into an area of geometry sometimes called “Power Theorems”. We will be dealing with three theorems that involve tangents, chords and secants and the measurement of segments of these figures.

We will need the properties of similar triangles for this. A future lesson!!

21

Given a circle C, and a point Q of its exterior. Let L 1 be a secant line through Q, intersecting C in points R and S; and let L 2 be another secant line through Q, intersecting C in points U and T. Then

QR · QS = QU · QT

S

R

Q

U

T

22

(4)

S

R

Q

U

T

What will we prove?

Given: Drawing

What is given?

Prove: QR · QS = QU · QT

(1) Q = Q

Reflexive

Why?

(2) QSU = QTR

Intercept same arcs.

Why?

(3) QSU ~ QTR

Why?

AA.

Why?

Property similar s.

Why?

Arithmetic.

(5) QR · QS = QU · QT

QED

23

Given a tangent segment QT to a circle, and a secant line through Q, intersecting the circle in points R and S. Then

QR · QS = QT 2

S

R

Q

T

24

S

R

Q

T

Given: Drawing

What is given?

What will we prove?

Prove: QR · QS = QT 2

For Homework.

Prove QST ~ QTR and set up the appropriate proportion to cross multiply to get QR · QS = QT 2

25

Let RS and TU be chords of the same circle, intersecting at Q. Then

QR · QS = QU · QT

S

Q

T

U

R

26

S

Q

T

U

R

Given: Drawing

What is given?

Prove: QR · QS = QU · QT

What will we prove?

For Homework.

Prove SQU ~ TQR and set up the appropriate proportion to cross multiply to get QR · QS = QU · QT.

27

What is given?

What will we prove?

Given:

Prove:

(1) Statement 1

Why?

Reason 1.

(2) Statement 2

Why?

Reason 2.

(3) Statement 3

Reason 3.

Why?

(4) Statement 4

Why?

Reason 4.

(5) Statement 5

Reason 5.

Why?

(6) Statement 6

Why?

Reason 6.

(7) Statement 7

Why?

Reason 7.

(8) Statement 8

Why?

Reason 8.

QED

DRAWING

30

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