1. C HAPTER 10 10.1 Circles and Circumference 10.2 Angles and Arcs 10.3 Arcs and Chords 10.4 Inscribed Angles 10.5 Tangents 10.6 Secants, Tangents, and Angle Measures 10.7 Special Segments in a Circle 10.8 Equations of Circles

2. 10.1 Circles and Circumference Objective: We will be able to identify and use the parts of a circle, including the circumference. A B L P Q Infinitely many chords, radii, and diameters Activity M Circle: All points in a plane equidistant from a given point. Diameter: Chord that passes through the center. Radius: Segment with center of circle and edge of circle as end points. Chord: Segment with endpoints connecting 2 points on a circle. Diameter = Circumference * = = 2r r *distance around the circle

3. N L C P Example 1: C is the center M Name: Diameters: Chords: Radii: N P R M Example 2: ST and QN are diameters Q If ST = 18, RS = If RN = 2, RP= If RM =24, QN= T S Example 4: Example 6: Find the EXACT circumference of circle P. Example 5: Find the EXACT circumference of the circle. a) If r = 13in., C = __________________ b) If d = 6mm, C = __________________ c) If C = 65.4 ft., d = ______, r = ______ E X F Z Y 9 mm P 9 m 5 m (No rounding!)

4. Find the EXACT circumference of circle P. P 15 m T S W Warm-Up: 20.1 units 3x 25x P 2x H L S V

5. 10.2 Angles and Arcs S Q 1 Objective: We will be able to recognize major arcs, minor arcs, semicircles, and central angles and their measures, as well as find arc length. 3 2 T (13x-3) (8x-4) (5x+5) (20x) V U R Minor arc Major arc Semicircle B C A B C A B C A D D central angles

6. R Q P S Adjacent Arcs - have exactly one point in common + = Arc Addition Postulate M K P N L O J Arc Length = l circumference degree measure of the arc R Q P degree measure of whole circle Remember! Arc measure is an angle : degrees Arc length is a distance: units

7. 10.3 Arcs and Chords Objective: We will be able to recognize and use relationships between arcs and chords, as well as between chords and diameters. D E D B A Theorem - then A polygon whose vertices all lie on a circle can be C Since Chords of adjacent arcs can form a polygon: V B A Theorem - T C Polygon RSTU is in circle C. inscribed Arc of a Chord: is the arc of chord S R U T Example 1: A regular octagon is inscribed in a circle. Find the number of degrees in each arc. in a circle, or circumscribed if the sides are tangent to a circle. C U about a circle,

8. H C M L H W K J b. Find JL A M H R O T Theorem - F E G H R Q P Y X P if PX = PY then,

9. 10.4 Inscribed Angles Objective: We will be able to find measures of inscribed angles, and measures of angles of inscribed polygons. : has a vertex on the circle & sides contained in chords within the circle. An inscribed angle D B E A C Theorem: Inscribed Angle = the measure of the intercepted arc. E D F T B T A C = = 2 ( ) S 1 2 O intercepted

10. Theorem: If an inscribed angle intercepts a ___________, the angle is a _____________. O Theorem: If two inscribed angles intercept congruent arcs or the same arc, then the angles are __________ B A C Q 3 4 1 2 D E Q S A T Theorem: If a quadrilateral is inscribed in a circle, opposite angles in a quad. are supplementary. 2 3 R P 2 3 1 4 1 1 4 ____ + ____ = ____ Example 5: 2 semicircle right angle

11. Central Angles. Vertex is at the – sides formed by two ____________. Warm-Up: 1 A B Inscribed Angles. Vertex is _________________ – sides formed by two _____________. A B C 1 A B 1 A B C 1 160 center on the circlechords radii

12. 10.5 Tangents Objective: We will be able to use the properties of tangents and solve problems involving circumscribed polygons. D E _________________: a line that intersects a circle at exactly _________________. Theorem: If a line is tangent to a circle, it is ____________________ to the radius drawn to the point of tangency. O Converse: If a line is perpendicular to a radius, then _________________ _________________ ________________. T R tangent to the line is perpendicular the circle. R Q A C 8 10 C B 7 9 7 7 Tangent one point 8 2 + x 2 = 10 2 64 + x 2 = 100 x 2 = 36 x = 6 AR = 12 9 2 + 7 2 ?=? 14 2 81 + 49 ?=? 196

13. E W 10 24 16 D Draw two lines tangent to circle V from point W. Theorem: If two segments from the same exterior point are tangent to a circle then, __________________________. C B A V H G E N J M Example 4: Assume the lines are tangent to both circles. Find x and y. W they are congruent F S T 10 y x + 4 L G K H 16 18 Yes! y = 10 x = 1 P = 158

14. Warm-Up: 23

15. 10.6 Secants, Tangents, and Angle Measures Objective: We will be able to find the measures of angles formed by lines intersecting on or inside a circle, and outside the circle. D E ____________: a line that intersects a circle at exactly ________________. Secant two points F Inside see ex. 1 Edge see ex. 2 A Outside see ex. 3 see ex. 4 see ex. 5 B D C 1 2 Two ________ intersect in the _________the circle. A ___________ and a _____________ intersect at the point of tangency. Two __________ intersect on the __________ of a circle. A __________ and a _____________ intersect on the ________________ of the circle. Two ___________ intersect on the _________ of the circle. C A B D E C A B D E C A B D C A B D secants interior secant tangent tangents exterior secant tangent exterior secants exterior

16. Example 1: Secant-Secant or chord-chord (inside) 3 2 4 Example 2: Secant-Tangent (edge) Example 3: Secant-Secant (outside) Example 5: Tangent-Tangent (outside) Example 4: Secant-Tangent (outside) 45 + 75 2 = 60 88 + 76 2 = 82 180 – 82 = 98 360 – 114 – 136 = 110 2 110 55 =

17. AE BE EC ED C B A D E = For lesson 10.7

18. 10.7 Special Segments in a Circle Objective: We will be able to find the measures of segments that intersect in the interior and exterior of a circle. Theorem: T wo intersecting cords C B A EC D E AE BE ED C B A AC D E AB AD AE Theorem: T wo secants to an exterior point. Example 1: x a) Find x : 12 9 8 x 3 9 6 b) Find x : Example 2: a) Find x : b) Find x : x 10 24 8 x 10 5 3 8 3 x x = 13.5 x = 2 x = 34.5 x = 2 segment outside whole segment segment outside whole segment AE BE EC ED = x = 12 8x = 12(9) 8x = 108 9x = 3(6) 9x = 18 8(8+x) =10(10+24) x(x+10) = 3(3+5) 10(8+10) = x(x+3)

19. Example 3: Z X Y XY W YZ YW Theorem: T angent and Secant to an exterior point. segment outside whole segment = segment outside whole segment x a) Find x x = 8 x + 4 x + 2 20 b) Find x x = 28.1 31 x *Same with 2 tangents

20. 10.8 Equations of Circles Objective: We will be able to write the equation of a circle and graph circles on the coordinate plane. All points in a plane equidistant from a given point (the center) * Draw a circle with center (3,2) and radius 4. * Every point on the circle is ___ units away from (3, 2). The distance from (3, 2) to (x, y) is 4. (x,y) So… the equation of a circle with center (h, k) and radius r is: (x,y) (h,k) r radius 2 “x” of center “y” of center r (3,2) 4 * note: opposite signs for the center ! square both sides to get Circle:

21. (-1,2) (1, -3) (0, 0) Example 4: Write an equation for each circle. a) Center at (-4, 2) and a radius of 2 b) Center at (0, 5) and a radius of 10 c) Center at (0, 0) and contains point (0, -3)