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Transcript
CC Geometry 10H Aim #1: What is the relationship between radii and tangents of circles? Do Now: 1. If the bicycle wheel travels 63 in. in one complete revolution 0 and rotates only 120 about the center, how far does it travel? 2. If a wheel spoke measures 18 in, what is the circumference and area of the wheel, to the nearest inch? center Definition: A circle is a set of points equidistant from a fixed point, called the _______. a segment from the center to any point on the circle Definition: A radius is___________________________________________________. • All radii of a circle are congruent. A O B D C circles whose diameters (or radii) are equal Definition: Equal circles are _______________________________________. congruent • All radii of equal circles are ___________. A B circle A = circle B a segment connecting 2 points on the circle's circumference Definition: A chord is_____________________________________________. a chord that passes through the center of the circle Definition: A diameter is________________________________________________. chord radius diameter twice • The length of a diameter of a circle is _______________ the length of a radius. Definition: An arc is a portion of the circumference of a circle. diameter of a circle. Definition: A semi-circle is an arc whose endpoints form a _________ than greater a semi-circle. ________ Definition: A major arc is an arc that is ___________ minor arc is an arc that is less than a semi-circle. Definition: A ___________ Definition:A secant is__________________________________________________. a line that intersects the circle at 2 points a line (seg/ray) that intersects the circle at exactly one point Definition: A tangent is________________________________________________. where the tangent line and circle intersect Definition: A point of tangency is__________________________________________. tangent secant point of tangency Name each figure: D E C A O diameter center _____ radius _____ _____ _____ secant ______ F B H chord _____ _____ G I _____ tangent _____ point of tangency ____ J A tangent is perpendicular to a radius drawn to the point of THEOREM: ____________________________________________________________ tangency. of tangency _______________________________________________________________________ A P B P = point of tangency O If a line is perpendicular to a radius at the point of CONVERSE of THEOREM _______________________________________________ intersection on circle, it is a tangent. ______________________________________________________________________ Exercises : Assume the lines that appear tangent are tangent. 3. Find x. 2. Find x. 1. Find x. L x O O xo O 38 0 x 117 0 N 22 0 4. Is XY 6. Find x. 5. Find x. YV ? 24 Show work to justify your answer. 12 O X 8 4 Y V x 32 x 8 O M 8. Find x. 7. Find x. 9 x O O 8 560 x0 9. BC = 9, AB = 6, AC = 15. Is BC tangent to circle A? Explain. 10. x = segment joining the centers. Find x, to the nearest tenth. 9.3 .5 26 2.4 x 11. A belt fits tightly around two circular pulleys as shown. Find the distance between the centers of the pulleys, to the nearest tenth. 35 in. 14 in. 8 in. 12. Isosceles ΔABC is inscribed in circle O, and base AC is a diameter. AB = -3x + 61, BC = x - 3 and AC = (x - 3)√2. a) Draw a labeled diagram b) Find the perimeter of ΔABC c) Name the largest angle of ΔABC. d) Find the length of a radius of circle O. e) Name a semi-circle of circle O. f) Name a major arc of circle O. B C Let's Sum it Up! D O E A G F O = center = radii = diameter = chord, not a diameter = tangent C = point of tangency = secant = semi-circle = major arc = minor arc Name __________________ Date_________ CC Geometry H HW #1 1. Name, in symbolic form: C a) a diameter _________ F B b) a tangent _____ O A c) a chord _______ G d) a secant _______ D e) a right angle ______ E f) two complementary angles _____ _____ 2a) Find x. b) Find x. c) Find x ,to the nearest tenth. 15 7 24 O x 9 10 cm x 7 cm x 3. In circle O, diameter AB = 3x + 11 and diameter CD = 5x - 23. Find the length of a radius. 4. In circle O, radius 2 OA = x + 16 and radius OB = 4x + 48. Find all possible lengths of a diameter. 5. D and E are points of tangency. Radius AE = 5 cm, CD = 12 cm. a) Find CA. D 12 cm A C 5 cm b) By what congruence criteria is ΔCDA ≅ ΔCEA? E c) What is true about m≮DCA and m≮ECA? Why? 6. Find x, the length of the common tangent segment between the two circles (nearest hundredth). x G 5 4 3 H Review 7. Find the perimeter of the triangle to the nearest tenth. 1050 300 10 8. The larger cone has a similar smaller cone with half its radius inside. What fraction of the larger cone's volume is outside the smaller cone? 9. If sin θ = , find cos θ and tan θ. THE "PERFECT" CIRCLE? http://www.youtube.com/watch?v=eAhfZUZiwSE What is the definition of a circle?