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Study Guide Unit 3 β Circles and Volume MCC9-12.G.C.1 Prove that all circles are similar. 1. Given a circle of a radius of 3 and another circle with a radius of 5, compare the ratios of the two radii, the two diameters, and the two circumferences. 2. What is the ratio of circumference to diameter of any circle? ___________ MCC9-12.G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. 3. In circle P, πβ π΄ππ΅ = 80°. Μ = _____ mπ΄π΅ Μ = _____ mπ΄πΆπ΅ 4. In the figure on the right, πβ π΅ππΆ = 85°, πβ π΄ππ· = Μ = 80° 120°, and ππΈπ· Find: Μ = _____ ππ΅πΆ Μ = _____ ππ΄π· πβ π΄π΅π· = _____ πβ πΈπ΅π· = _____ Μ = 85°. 5. β πππ is inscribed in circle O and πππ a. What is the measure of β πππ? 6. In circle P, Μ Μ Μ Μ π΄π΅ is a diameter. If πβ π΄ππΆ = 94°, find the following: a. πβ π΅ππΆ b. What is the measure of β πππ ? Μ b. ππ΅πΆ c. πβ π΅π΄πΆ Μ d. ππ΄πΆ 7. In this circle, Μ Μ Μ Μ π΄π΅ is tangent to the Μ Μ Μ Μ is tangent to circle at point B, π΄πΆ the circle at point C, and point D lies on the circle. What is πβ π΅π΄πΆ? 8. In the circle shown, Μ Μ Μ Μ π΅πΆ is a Μ diameter and ππ΄π΅ = 120°. What is β π΄π΅πΆ ? Μ Μ Μ Μ and ππ Μ Μ Μ Μ and chords ππ Μ Μ Μ Μ Μ 9. Circle P has tangents ππ Μ Μ Μ Μ Μ and ππ, as shown in this figure. The measure β πππ = 60° . 10. Solve for x. What is β πππ ? 11. Find the mβ ABD, the inscribed angle of βC. 13. Find the mβ ABD, the inscribed angle of βC. 15. In circle P, Μ Μ Μ Μ π·πΊ is a tangent. 12. Find the mβ ABD, the inscribed angle of βC, if Μ = 240° ππ΅πΈπ· 14. A diameter of a circle is perpendicular to a chord whose length is 12 inches. If the length of the shorter segment of the diameter is 4 inches, what is the length of the longer segment of the diameter? 16. Chords AB and CD intersect inside a circle at point E. AE= 2, CE =4 , and ED =3 . Find EB. AF = 8, EF = 6, BF = 4, and EG = 8. a. Find CF b. Find DG 17. Chords AB and CD intersect inside a circle at point E. AE= 5, CE =10, EB = x, and ED = x-4. 18. Two secant segments are drawn to a circle from a point outside the circle. The external segment of the first secant segment is 8 centimeters and its internal segment is 6 centimeters. If the entire length of the second secant segment is 28 centimeters, what is the length of its external segment? 19. A tangent segment and a secant segment are drawn 20. The diameter of a circle is 19 inches. If the to a circle from a point outside the circle. diameter is extended 5 inches beyond the circle The length of the tangent segment is 15 inches. The to point C, how long is the tangent segment from external segment of the secant segment measures 5 point C to the circle? inches. What is the measure of the internal secant segment? Find EB and ED. Page 2 21. A satellite orbits the earth so that it remains at the same point above the Earthβs surface as the Earth turns. a. If the satellite has a 50° view of the equator, what percent of the equator can be seen from the satellite? b. The average radius of the Earth is approximately 3959 miles. How far above the Earthβs surface is the satellite? c. What is the length of the longest line of sight from the satellite to the Earthβs surface? Identify this line of sight using the diagram. MCC9-12.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. 22. ABCD is an inscribed quadrilateral. If πβ πΆπ·π΄ = 130° and πβ π·π΄π΅ = 75°, thenβ¦ Find: πβ π΄π΅πΆ = ______ and πβ π΅πΆπ· = ______ MCC9-12.G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. 23. Complete the table and consider the ratio of arc length to radius for different radii. Angle Radius 45° 45° 45° 45° 2 4 9 13 Exact Arc Length Exact Arc Length / Radius What is the radian measure of 45°? What do you notice about the ratio of the arc length to radius? Μ ? Leave your answer in 25. What is the length of πΆπ· terms of pi. 24. If x = 50°, what is the area of the shaded sector of circle A? Leave your answer in terms of pi. Page 3 26. What is area of the shaded part of the circle below? Leave your answer in terms of pi. 27. The spokes of a bicycle wheel form 10 congruent central angles. The diameter of the circle formed by the outer edge of the wheel is 18 inches. What is the length, to the nearest 0.1 inch, of the outer edge of the wheel between two consecutive spokes? MCC9-12.G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieriβs principle, and informal limit arguments. MCC9-12.G.GMD.2 (+) Give an informal argument using Cavalieriβs principle for the formulas for the volume of a sphere and other solid figures. MCC9-12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.β 28. What is the volume of the cone shown below? 29. A cylinder has a radius of 10 cm and a height of 9 cm. A cone has a radius of 10 cm and a height of 9 cm. Show that the volume of the cylinder is three times the volume of the cone. 30. A sphere has a radius of 3 feet. What is the volume of the sphere? 31. What is the volume of a cylinder with a radius of 3 9 in. and a height of 2 in.? 32. Letβs investigate the relationship between a Cone and its corresponding Cylinder with the same height and radius. LABEL the height, h and radius, r on each diagram below. 33. Letβs investigate the relationship between a Pyramid and its corresponding Rectangular Prism with the same height, length, and width. LABEL the height, h, length, l, and width, w, on each diagram below. a. If the cylinder was full of water and you poured it into the cone, how many times would it fill up the cone completely? b. If the cone was full of water, how much of the cylinder would it fill up? Page 4 c. If the rectangular prism was full of water and you poured it into the pyramid, how many times would it fill up the pyramid completely? d. If the pyramid was full of water, how much of the rectangular prism would it fill up? 34. Approximate the Volume of the Backpack that is 17 in x 12 in x 4 in. 35. Find the Volume of the Grain Silo shown below that has a diameter of 20 ft and a height of 50 ft. 36. Cylinder A and cylinder B are shown below. What is the volume of each cylinder? 37. The volume of a cylindrical watering can is 100 cm3. If the radius is doubled, then how much water can the new can hold? 38. Jason constructed two cylinders using solid metal washers. The cylinders have the same height, but one of the cylinders is slanted as shown. Which statement is true about Jasonβs cylinders? A. The cylinders have different volumes because they have different radii. B. The cylinders have different volumes because they have different surface areas. C. The cylinders have the same volume because each of the washers has the same height. D. The cylinders have the same volume because they have the same cross-sectional area at every plane parallel to the bases. Page 5