Geo 9 1 Circles 9-1 Basic Terms associated with Circles and Spheres. Radius. Chord. Secant. Diameter. Tangent. Point of Tangency.
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1 Geo 9 1 ircles 9-1 asic Terms associated with ircles and Spheres ircle Given Point = Given distance = Radius hord Secant iameter Tangent Point of Tangenc Sphere Label ccordingl: ongruent circles or spheres oncentric ircles oncentric Spheres Inscribed in a circle/circumscribed about the polgon
2 Geo 9 2 ircles SKETHP
3 Geo 9 3 ircles 9-2 Tangents PWERPINT Theorem 9-1 If a line is tangent to a circle, then the line is. orollar: Tangents to a circle from a point are P Theorem 9-2 If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is. Inscribed in the polgon/circumscribed about the circle: look for 2 tangents from the same point! what if is a right ange?
4 Geo 9 4 ircles ommon Tangent ommon Internal Tangent ommon Eternal Tangent Tangent circles raw the tangent line for each drawing Name a line that satisfies the given description. F P 1. Tangent to P but not to. 2. ommon eternal tangent to and P. 3. ommon internal tangent to and P.
5 Geo 9 5 ircles 4. ircles,, are tangent. = 7, = 5 = 9 Find the radii of the circles. 5. Find the radius of the circle inscribed in a triangle PP NLUSIN
6 Geo 9 6 ircles 6) ircles and P have radii 18 and 8 respectivel. is tangent to both circles. Find.Hint: connect centers. Find a rt. P
7 Geo 9 7 ircles 9-3 rcs and entral ngles entral ngle rc Measure of a minor arc = Measure of a major arc = - djacent arcs Measure of a semicircle = Postulate 16 rc ddition Postulate: The measure of the arc formed b two adjacent arcs is. That is, arcs are additive. Just like with angles, to differentiate an arc from its measure, an m must be included in front of the arc. ongruent arcs Theorem 9-3 In the same circle or, two minor arcs are if. S R 1. Name 2. Give the measure of each angle or arc: a) two minor arcs a) b) two major arcs b) m WT c) a semicircle c) XYT d) an acute central angle e) two congruent arcs X W Y T Z
8 Geo 9 8 ircles 3. Find the measure of 1 (the central angle) 40 a) b) c) d) Find the measure of each arc: E 3 a) b) c) d) E e) E 5) a) If 60, = 10, find <1, <2 and 2 1 b) If <2 = find <1,
9 Geo 9 9 ircles 9-4 rcs and hords The arc of the chord is Theorem 9-5 diameter that is perpendicular to a chord the chord and. That is, in with, Z = Z and How? Z ther Theorems: If < = <, then what must be true as well? 1) 2) 3) 4)
10 Geo 9 10 ircles Find the following: 1. = = 2. = = m = MN = K = 4. = m = S M K 15 N = = 6. m = = 40, FIN 8. If = 6, find and E
11 Geo 9 11 ircles 9-5 Inscribed ngles definition, an inscribed angle is an angle whose VERTEX IS N THE IRLE and is contained in the circle. Inscribed angles can intercept a minor arc or a major arc. Theorem 9-7 The measure of an inscribed angle is equal to Find angle and angle. What generalization can ou make? 70 orollar 1: If two inscribed angles orollar 2: n inscribed angle that intercepts a diameter
12 Geo 9 12 ircles orollar 3: If a quadrilateral is inscribed in a circle, then its opposite angles are Y X Theorem 9-8 the measure of an angle formed b a chord and a tangent is equal to of the intercepted. Solve for the variable(s) listed: z z
13 Geo 9 13 ircles PWERPINT
14 Geo 9 14 ircles 9-6 ther ngles Sketchpad Theorem 9-9 The measure of an angle formed b two chords that intersect inside a circle is equal to 1 2 the sum of the intercepted arcs. That is: 1 Theorem 9-10 The measure of an angle formed b secants, two tangents or a secant and a tangent is equal to THE VERTEX IS UTSIE THE IRLE ase 1 ase 2 ase 3 2 secants 2 tangents secant/tangent
15 Geo 9 15 ircles Given UT is tangent to the circle, m VUT = 30. Find the following: U 100 W R T V 100 S 1. m WT = 2. m TVS = 3. m RVS = 4. m RS = Given the drawing: is tangent to ; F is a diameter; m G = 100, m E = 30, m EF = 25. Find the measures of angles = 2= E 3= F 4= 5= 6= 7= 8= G
16 Geo 9 16 ircles NGLE MESUREMENT SE N VERTEX 1) VERTEX T ENTER angle = 2) VERTEX N IRLE angle = 3) VERTEX INSIE IRLE angle = 4) VERTEX UTSIE THE IRLE angle = SENT/SENT TNGENT/SENT TNGENT/TNGENT
17 Geo 9 17 ircles 9-7 ircles and Lengths of Segments Theorem 9-11 When two intersect inside a circle, the of the of equals the of the of the. That is, in the circle below, given that the two chords intersect, the equation is r u t s or Theorem 9-12 When two segments are drawn to a circle from an, the product of one secant segment and its is equal to the product of the other secant segment and its That is, in the circle below, r or s u t Theorem 9-13 When a segment and a segment are drawn to a circle From an the product of the secant segment and Its is equal to the of the. That is, in the circle below: r or s t
18 Geo 9 18 ircles EXMPLES: SKETHP PWERPINT
19 Geo 9 19 ircles 40 Find the measure of each numbered angle given arc measures as indicated is a central angle m 1 m 2 m 3 m 4 m 5 50 m 6 m 7 m 8 m 9 m 10 m 11 m 12 m 13 m 14 m 15 m 16 m 17 m 18 m 19 m 20 m 21 m 22 m 23 m 24 m 25 m 26 m 27 m 28 m 29 m 30 m 31 m 32 m 33 m 34 m 35 m 36 m 37 m 38 m 39 m 40 m 41 m 42 m 43 m 44 m 45
20 Geo 9 20 ircles H 9 IRLE REVIEW (1) Find the measure of each of the numbered (2) The three circles with centers,, and angles, given the figure below with arc are tangent to each other as shown below. measures as marked. Point is the center Find the radius of each circle if = 12, of the circle. 60 = 10 and = m 1 = m 2 = m 3 = m 4 = m 5 = m 6 = m 7 = m 8 = ircle, ircle, ircle m 9 = m 10 = (3) m = 120, = 6. Find: (4) m = 80 Find: m (5) is tangent to the circle with center. (6) is a diameter,, = 3, = 2, = 3. Find: = 6. Find:
21 Geo 9 21 ircles (7) E is tangent at, is a diameter, (8) is a diameter, is tangent at, m = 40. Find: m, m E m = 120, = 6 3. E Find:,, (9) is tangent at, F = F, sides as marked. (10) Given the figure with sides as marked, Find: EF, F Find:, EF E F E F (11) ircles with centers and P as shown, (12) Given the figure below with sides as P = 15, = 8, P = 4 marked, find the radius of the inscribed Find:, circle P 12 F 20 E 16
22 Geo 9 22 ircles nswers (1) m 1 = 20, m 2 = 25, m 3 = 55, m 4 = 90 m 5 = 25, m 6 = 115, m 7 = 65, m 8 = 115 m 9 = 45, m 10 = 130 (2) ircle = 7, ircle = 5, ircle = 3 (3) 6 3 (4) m = 260 (5) = 4 (6) = 3 2 (7) m = 130, m E = 65 (8) = 4 3, = 2 3, = 6 (9) EF = 9, F = 6 (10) = 4, EF = 8 (11) = 9, = 209 (12) 4
23 Geo 9 23 ircles H 9 IRLES REVIEW II (1) The circle with center is inscribed in. (2) is tangent to the circle at, sides as. Find:, marked. Find: 4 F E (3) is an eternal tangent segment. Points (4) oncentric circles with center, is and P are the centers of the circles. tangent to the inner circle, sides as marked. Find: Find:, m P 4 8 (5) Given the figure below, point is the center (6) Given the figure below, m = 30, the circle,, = 26, = 24. m F = 65, = E. Find: E, E, Find: m, me, m E 65 F 30 E
24 Geo 9 24 ircles (7) The circle below with center, = 12, (8) Given the figure below, H = HF, with. sides as marked. Find: E, E Find: G, H E 3 4 G 6 H E F (9) The circle with center is inscribed (10) Points and P are the centers of the in as shown below. =, circles below. P = 6 sides as marked. Find: E Find:, m 8 P 6 E 5 F (11) chord whose length is 30 is in a circle whose radius is 17. How far is the chord from the center of the circle?
25 Geo 9 25 ircles Review nswers II (1) = 6, = 8 (2) = 6 3 (3) = 4 6 (4) = 4, m = 240 (5) E = 5, E = 8, = 13 (6) m = 95, me = 35, m = 115 (7) E = 2 3, = 4 3, E = 2 3 (8) G = 27 4, H = 3 3 (9) E = 10 3 (10) = 6 3, m = 240 (11) 8
26 Geo 9 26 ircles H 9 IRLES ITINL REVIEW 1) Find the radius of a circle in which a 48 cm chord is 8 cm closer to the center than a 40 cm chord. = 48, = 40 2) In a circle, PQ = 4 RQ = 10 P = 15. Find PS. R Q P S 3) n isosceles triangle, with legs = 13, is inscribed in a circle. If the altitude to the base of the triangle is = 5, find the radius of the circle. (There are 2 situations) nswers: 1) 25 2) 2 3) 16.9
27 9.3 RS N ENTRL NGLES 9.2 TNGENTS Geo 9 27 ircles 1) Fill out page one of the ircles Packet. SUPPLEMENTRY PRLEMS H 9 2) regular polgon is inscribed in a circle so that all vertices of the quadrilateral intersect the circle. What happens to the regular polgon as the number of sides increases. 3) circle with a center at (2,1) is tangent to the line = at (-1,2). Make a sketch in the coordinate plane and draw a radius from the center of the circle to the radius at point? Wh? 4) In the picture below, is a common eternal tangent. How man common eternal tangents can be drawn connecting the 2 circles in each of the following pictures? What shape can be formed if a radius drawn to a tangent is perpendicular to the tangent? 5) If the central angle of a slice of pizza is 36 degrees, how man pieces are in the pizza? 6) ircle has a diameter G and central angles G = 86, E = 25, and FG = 15. Find the minor arcs G, F, EF, and major arc GF. 7) raw a circle and label one of its diameters. hoose an other point on the circle and call it. What can ou sa about the size of angle? oes it depend on which ou chose? Justif our response, please.
28 9.5 INSRIE NGLES 9.4 RS N HRS Geo 9 28 ircles 8) If two chords in the same circle have the same length, then their minor arcs have the same length, too. True or false? Eplain. What about the converse of the statement? Is it true? Wh? 9) raw a circle. raw two chords of unequal length. Which chord is closer to the center of the circle? What can be said of the intercepted arcs? 10) If P and Q are points on a circle, then the center of the circle must be on the perpendicular bisector of chord PQ. Eplain. Which point on the chord is closest to the center? P Q 11) The Star Trek Theorem: a.) Given a circle centered at, let,,and be points on the circle such that arc is not equal to arc and L is a diameter. Wh must triangles and be isosceles? b) State the pairs of angles that must be congruent in these isosceles triangles. c) Using ET, find epressions for the measures of <L and <L. d) ased on our statement in part c, eplain the statement <L = ½(<L) and < = ½(<L). e) Now find an epression for < and simplif to prove that it equals ½<. L
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