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**Lesson 10.1 Parts of a Circle Today, we are going to…**

> identify segments and lines related to circles > use properties of tangents to a circle

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Circle C C Diameter = _ radius

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A chord is X Y N YX C A B AB BN

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A secant is X Y C A B YX AB

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A tangent is C Y X XY AB A B

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Common Tangent Lines internal tangents

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Common Tangent Lines external tangents

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**2 points of intersection**

Two circles can intersect in 2, 1, or 0 points. Draw 2 circles that have 2 points of intersection

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**internally tangent circles**

Draw two circles that have 1 point of intersection

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**externally tangent circles**

Draw two circles that have 1 point of intersection

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**no point of intersection**

concentric circles Draw two circles that have no point of intersection

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**9. What are the center and radius of circle A?**

Center: Radius =

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**10. What are the center and radius of circle B?**

Center: Radius =

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**11. Identify the intersection of the two circles.**

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**12. Identify all common tangents of the**

two circles.

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m Ð ABC = A B C

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Theorem 10.1 & 10.2 A line is tangent to a circle if and only if it is _____________ to the radius from the point of tangency. A B C

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13. Find CA. C 7 D B 15 What is DA? A

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14. Find x. C x 7 x x What is CA? B 8 6 16 15 A

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15. Is AB a tangent? How do we test if 3 segments create a right triangle? C 10 7 26 B 6 24 15 A

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16. Is AB a tangent? C 8 7 17 B 6 12 15 A

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**17. Find the slope of line t. A (3,0) and C (5, -1) t A Slope of AC? C**

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**One endpoint is the point of tangency.**

A tangent segment A B C One endpoint is the point of tangency.

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Theorem 10.3 If 2 segments from the same point outside a circle are tangent to the circle, then they are congruent.

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18. Find x. B 7x - 2 A C 3x + 8

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19. Find x. B x2 + 25 A 50 C

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**Lesson 10.2 Arcs and Chords Today, we are going to…**

> use properties of arcs and chords of circles

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**An angle whose vertex is the center of a circle is a**

central angle. C A B

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Minor Arc - Major Arc Major Arc ADB C D Minor Arc AB A B

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Measures of Arcs C A B D 60˚ m AB =

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Semicircle A B D E C m AED = m ABD = m AD

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**1. m BD 2. m DE 3. m FC 4. m BFD Find the measures of the arcs. D C**

68˚ 52˚ ? B 100˚ E 53˚ F

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**AD and EB are diameters. 5. Find x, y, and z. E F D C A B x = x˚ 30˚**

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**if and only if their chords are congruent.**

Theorem 10.4 Two arcs are congruent if and only if their chords are congruent.

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6. Find m AB B (3x + 11)° (2x + 48)° C D A

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Theorem 10.5 & 10.6 A chord is a diameter if and only if it is a perpendicular bisector of a chord and bisects its arc.

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7. Is AB a diameter? A B

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8. Is AB a diameter? A B 8

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9. Is AB a diameter? A B

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Theorem 10.7 Two chords are congruent if and only if they are equidistant from the center.

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10. Find CG. AB = 12 D G B A C F E DE = 12 x 7 6 ?

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**Lesson 10.3 Inscribed Angles Today, we are ALSO going to…**

> use properties of inscribed angles to solve problems

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An inscribed angle is an angle whose vertex is on the circle and whose sides contain chords of the circle.

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Theorem 10.8 If an angle is inscribed, then its measure is half the measure of its intercepted arc. 2x x

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1. Find x. x = 60° 120° x°

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2. Find x. x = 140° x° 70°

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Theorem 10.9 If 2 inscribed angles intercept the same arc, then the angles are congruent.

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3. Find x and y. x° 45° y°

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Inscribed Pentagon

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4. DC is a diameter. Find x. C A D x° B

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Theorem 10.10 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.

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**5. Find the values of x and y.**

C A y° 42 D x° B

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**Theorem 10.11 If a quadrilateral is inscribed in a circle, then its**

opposite angles are supplementary. 2 1 4 3 m 1 + m 3 = 180º m 2 + m 4 = 180º

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**6. Find the values of x and y.**

80° y° 110°

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**7. Find the values of x and y.**

100° y° 120°

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**Angle Relationships in Circles**

Lesson 10.4 Angle Relationships in Circles Today, we are going to… > use angles formed by tangents and chords to solve problems > use angles formed by intersecting lines to solve problems

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**If a tangent and a chord intersect at a point on a circle, then...**

Theorem 10.12 If a tangent and a chord intersect at a point on a circle, then... GSP

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**… the measure of each angle formed is half**

Theorem 10.12 … the measure of each angle formed is half the measure of its intercepted arc.

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B C 2 1 A

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1. Find m 1 and m 2. B C 100° 2 1 A

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2. Find and mACB and mAB 95° A B C

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5x° A B C (9x + 20)˚ 3. Find x

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**If 2 chords intersect inside a circle, then…**

Theorem 10.13 If 2 chords intersect inside a circle, then… A C 1 B D

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**…the measure of the angle is half the sum of the intercepted arcs.**

1 B D …the measure of the angle is half the sum of the intercepted arcs.

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4. Find x. 100° A B C D x° 120°

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5. Find x. 130° A B C D x° 160°

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6. Find x. A C x° 80° y° 90° B D

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7. Find x. x° A B C D 100° 120°

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8. Find x. A B C D 74° 52° x° Do you notice a pattern?

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**If a tangent and a secant, **

Theorem 10.14 If a tangent and a secant, two tangents, or two secants intersect outside a circle, then… A C D 1

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**If a tangent and a secant, **

Theorem 10.14 If a tangent and a secant, two tangents, or two secants intersect outside a circle, then… A B C 1

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**If a tangent and a secant, **

Theorem 10.14 If a tangent and a secant, two tangents, or two secants intersect outside a circle, then… A B C D 1

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A B C D 1 …the measure of the angle is half the difference of the intercepted arcs.

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9. Find x. A B C D x° 20° 80°

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10. Find x. 24° 90° A B C D x°

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11. Find x. x° 200°

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12. Find x. A C D 135° x°

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13. Find x. 100° 2 3 60° 100° 1 100°

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**Lesson 10.5 Segment Lengths in Circles**

Today, we are going to… > find the lengths of segments of chords, tangents, and secants

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Theorem 10.15 If 2 chords intersect inside a circle, then the product of their “segments” are equal.

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a c d b a · b = c · d

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1. Find x. 6 x 8 4

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2. Find x. 3x 3 18 2x

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3. Find x. 2x 18 x 4

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**Theorem 10.16 If 2 secant segments share the same endpoint outside a circle, then… GSP**

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**…one secant segment times its external part equals the other secant segment times its external part.**

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c a b d a · c = b · d

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5 x 4 6 3. Find x.

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9 10 x 20 4. Find x.

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**Theorem 10.17 If a secant segment and a tangent segment share an endpoint outside a circle, then…**

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…the length of the tangent segment squared equals the length of the secant segment times its external part.

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a b d a · a = b · d a2 = b · d

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5 4 x 5. Find x.

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15 x 10 6. Find x.

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♫ ♪ Quadratic Formula?

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15 x 10 6. Find x.

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x 20 31 7. Find x.

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3 4 8 x 8. Find x.

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10 x 8 9. Find x.

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**Lesson 10.6 Equations of Circles Today, we are going to…**

> write the equation of a circle

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Standard Equation for a Circle with Center: (0,0) Radius = r

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**1. Write an equation of the circle.**

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**2. Write an equation of the circle.**

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Standard Equation for a Circle with Center: (h,k) Radius = r

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**3.Write an equation of the circle.**

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**4.Write an equation of the circle.**

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Graph (x – 3)2 + (y + 2)2 = 9 Center? Radius =

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**Identify the center and radius of**

the circle with the given equation. 5. (x – 1)2 + (y + 3)2 = 100 Center: (1, -3) radius = 10 6. x2 + (y - 7)2 = 8 Center: (0, 7) radius ≈ 2.83 7. (x + 1)2 + y2 = ¼ radius = ½ Center: (-1, 0)

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Write the standard equation of the circle with a center of (5, -1) if a point on the circle is (1,2).

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**8. Write the standard equation of. the circle with a center of (-3, 4)**

8. Write the standard equation of the circle with a center of (-3, 4) if a point on the circle is (2,-5).

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Is (-2,-10) on the circle (x + 5)2 + (y + 6)2 = 25?

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9. Is (0, - 6) on the circle (x + 5)2 + (y – 5)2 = 169?

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10. Is (2, 5) on the circle (x – 7)2 + (y + 5)2 = 121?

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< > =

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**Would the point be inside the circle, outside the circle, or on the circle?**

(x – 13)2 + (y - 4)2 = 100 11. (11, 13) 12. (6, -5) 13. (19, - 4)

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**Circumference and Area of Circles**

Lessons 11.4 & 11.5 Circumference and Area of Circles Today, we are going to… > find the length around part of a circle and find the area of part of a circle

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Circumference

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Arc Length = A B

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1. Find the length of AB A B 50° 7 cm

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2. Find the radius A 10 cm 85° B

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3. Find the circumference.

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Area

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**A region bound by two radii & their intercepted arc.**

Sector of a circle A region bound by two radii & their intercepted arc. A slice of pizza!

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Area of a Sector =

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**3. Find the area of the sector.**

50° B 7 cm

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A B 100° 4. Find the radius.

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3. Find the area.

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Workbook P. 211 (1 – 10) P. 215 (1 – 6)

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