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Check your homework assignment with your partner!

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

A

B

C

QP

R16

2012

WARM UP EXERCSE∆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!

A

B

C

QP

R16

2012 x

20 - x

§10.1 Circles

The student will learn:

More about of circles and the lines associated with them.

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Parallel Lines and Circles

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.

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Parallel Lines and Circles

Theorem: Parallel lines intercept equal arcs on a circle.

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

C

BA

D

C

BA

DC

BA

D

Tangent-Secant Proof

Statement Reason

1. OE AB. Radius to tangent

Given: AB ‖ CD and AB tangent at E.

Prove: arc CE = arc DE

3. AB CD Given

4. OPD = 90 Corresponding angles

5. OE CD Def perpendicular

2. OEB = 90 Def perpendicular

Prove two secant case for homework.

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6. arc CE = arc DE to chord bisects arc

O

E

DC

BA

P

The other cases can be reduced to this case.

Think!

Non Parallel Lines and Circles

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.

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

Inscribed Angles (SS on C)Angle ABC is an inscribed angle of a circle if AB and BC are chords of the circle.

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

C

B

A

O

C

B

A

O

C

BA

O

Proof

Statement Reason

1. OB = OC. Radii

Given: Inscribed angle ABC .

Prove: ABC = ½ arc AC

Case 1: O on angle.

3. AOC = ABC + OCB Exterior angle thm.

4. 2ABC = AOC = arc AC Arithmetic

5. ABC = ½ arc AC Arithmetic

2. ABC = OCB Base angle isosceles ∆.

Prove other two cases for homework.

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C

B

A

O

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

Theorem (ST on C)An angle formed by a chord and a tangent at one end of the chord is half the intercepted arc.

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D C

BA

Proof

Statement Reason

1. Draw CD ‖ AB Construction

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

3. C = CAB Alternate interior angles

4. Arc AD = arc AC ‖ Lines intercept = arcs

5. CAB = ½ arc AC Substitution

2. C = ½ arc AD Inscribed angle

QED

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D C

BA

Non Parallel Lines and Circles

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

Non Parallel Lines and Circles

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

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

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E

D

CB

A

Proof

Statement Reason

1. AEC = A + D . Exterior angle thm.

Given: Chords AB and CD met at E.

Prove AEC = ½ (arc AC + arc BD)

3. D = ½ arc AC Inscribed angle

4. AEC = ½ (arc AC + arc BD) Substitution

2. A = ½ arc BD Inscribed angle

QED

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E

D

CB

A

Non Parallel Lines and Circles

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

Non Parallel Lines and Circles

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

TheoremAn 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

Statement Reason

1. AEC = A + B. Exterior angle theorem

Given: Secants AB & CB.

Prove: B = ½ arc AC - ½ arc DE

3. AEC = ½ arc AC Inscribed angle.

4. A = ½ arc DE Inscribed angle.

5. B = ½ arc AC - ½ arc DE. Substitution

2. B = AEC - A. Subtraction.

Prove other two cases for homework.

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E

D

C

BA

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.

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

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

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Q

TU

S

R

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

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Given: Drawing Prove: QR · QS = QU · QT

(1) Q = Q Reflexive

(2) QSU = QTR Intercept same arcs.

(3) QSU ~ QTR AA.

Property similar s.

What is given? What will we prove?

Why?

Why?

Why?

Why?

QED

(5) QR · QS = QU · QT Why?Arithmetic.

Q

TU

S

R

(4)QS QU

QT QR

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

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Q

T

S

R

The Tangent - Secant Power Theorem.

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Given: Drawing 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

What is given? What will we prove?

Q

T

S

R

The Two-Chord Power Theorem.Let RS and TU be chords of the same circle, intersecting at Q. Then

QR · QS = QU · QT

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Q

R

ST

U

The Two-Chord Power Theorem.

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Given: Drawing Prove: QR · QS = QU · QT For Homework.

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

What is given? What will we prove?

Q

R

ST

U

Assignment: §10.1

The Two-Chord Power Theorem.

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Given: Prove:

(1) Statement 1 Reason 1.(2) Statement 2 Reason 2.(3) Statement 3 Reason 3.(4) Statement 4 Reason 4.

What is given? What will we prove?

Why?Why?Why?Why?

QED

(5) Statement 5 Why?Reason 5.(6) Statement 6 Why?Reason 6.

DRAWING

(7) Statement 7 Why?Reason 7.(8) Statement 8 Why?Reason 8.