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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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C Circle C Diameter = _ radius

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

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C A secant is A B X Y YX AB

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C A tangent is AB A B Y X XY

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internal tangents Common Tangent Lines

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external tangents

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

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

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

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

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

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

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x 2 + 25 50 19. Find x. A C B

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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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C An angle whose vertex is the center of a circle is a central angle. A B

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

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C A B D 60˚ m AB = Measures of Arcs

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

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Find the measures of the arcs. 1. m BD 2. m DE 3. m FC 4. m BFD D E F B C 100˚ 52˚ 68˚ 53˚ ?

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

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

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

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

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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. xx 2x

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1. Find x. x°x° 120° x = 60°

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2. Find x. x°x° 70° x = 140°

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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. y°y° 45° x°x°

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Inscribed Pentagon

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

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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. x°x° y°y° A 42 D C 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. x°x° 110° 80° y°y°

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7. Find the values of x and y. x°x° 120° 100° y°y°

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

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

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1 A B C 2

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1 A B C 2 1. Find m 1 and m 2. 100°

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2. Find and mACB and mAB 95° A B C

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3. Find x 5x° A B C (9x + 20)˚

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

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

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A B C D x°x° 4. Find x. 100° 120°

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A B C D x°x° 5. Find x. 130° 160°

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A B C D x°x° 6. Find x. 80° 90° y°y°

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A B C D x°x° 7. Find x. 100° 120°

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

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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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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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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. 20° 80° A B C D x°x°

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10. Find x. 24° 90° A B C D x°x°

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11. Find x. 200° x°x°

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A C D 12. Find x. 135° x°x°

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13. Find x. 100° 3 2 1 60°

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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 · b = c · da · b = c · d a b c d

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1. Find x. 6 8 4 x

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2. Find x. 3x 18 2x 3

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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 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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a · c = b · d b a c d

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3. Find x. 5 x 4 6

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4. Find x. 9 10 x 20

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Theorem 10.17 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 · a = b · d d b a a2 = b · da2 = 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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8. Find x. 3 4 8 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. C = r =

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4.Write an equation of the circle. C = r =

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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 6. x 2 + (y - 7) 2 = 8 7. (x + 1) 2 + y 2 = ¼ Center: (1, -3)radius = 10 Center: (0, 7)radius ≈ 2.83 Center: (-1, 0) radius = ½

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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) 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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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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A B 50° 7 cm 1. Find the length of AB

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A B 85° 10 cm 2. Find the radius

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3. Find the circumference.

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Area

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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. A B 50° 7 cm

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4. Find the radius. A B 100°

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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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