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Check your homework assignment with your partner!. 1. 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.

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Presentation Transcript
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

§10.1 Circles

The student will learn:

More about of circles and the lines associated with them.

4

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 Circles

5

Parallel Lines and Circles

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 Circles

Let’s begin with the case where the lines intersect on the circle.

8

Inscribed Angles (SS on C)

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:

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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

Theorem (ST on C)

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

D

C

B

A

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Proof

D

C

B

A

Given: Chord AC, tangent AB Prove: CAB= ½ arc AC

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Non Parallel Lines and Circles

There is no case of two tangents intersecting on the circle.

13

Non Parallel Lines and Circles

We now move to the cases where the lines meet inside the circle.

14

Theorem

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

C

E

B

A

D

15

Proof

C

E

B

A

D

Given: Chords AB and CD met at E.

Prove AEC = ½ (arc AC + arc BD)

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Non Parallel Lines and Circles

There is no cases involving tangents meeting inside of the circle.

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Non Parallel Lines and Circles

And finally to the cases where the lines meet outside the circle.

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Theorem

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

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Two Secant Proof

A

D

B

Given: Secants AB & CB.

Prove:  B = ½ arc AC - ½ arc DE

E

C

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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!!

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The Two-Secant Power Theorem.

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

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

(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

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The Tangent - Secant Power Theorem.

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

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The Tangent - Secant Power Theorem.

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

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The Two-Chord Power Theorem.

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

QR · QS = QU · QT

S

Q

T

U

R

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The Two-Chord Power Theorem.

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.

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The Two-Chord Power Theorem.

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

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