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Chapter 6 Similarity Ratio is a comparison between 2 “things” A proportion is two equal ratios Similar Polygons o Corresponding angles are congruent o Corresponding sides are proportional Triangle Similarity o AA angle-angle similarity, SSS side-side-side similarity, SAS side-angle-side similarity Parallel lines can intersect 2 or more transversals proportionally Exercises for Similarity: 1.______ Assuming the triangle below are similar, find the height of the building. 2. ____ ΔPGT~ΔRST, what is the length of QT?, If PT = 15, what is the ______ length of RT? 3. a)_____ A soccer team played 24 games and won 16 games. a) What is the ratio of the number of wins b)_____ to loses b) What is the ratio of the number of games played to games won? For each pair of figures, determine if there is a similarity transformation that maps one figure onto the other. 4. ______ 5._______ 6. ________ 7 7 7 Chapter 7: Quadrilaterals and Polygons Identify special quadrilaterals and use their properties to solve problems. o Parallelogram Both pairs of opposite sides parallel and congruent () Both pairs of opposite angles congruent, and consecutive angles are supplementary Diagonals bisect each other o Rectangle All the properties of a parallelogram 4 right angles Diagonals are congruent o Rhombus All the properties of a parallelogram 4 congruent sides Diagonals are perpendicular, and diagonals bisect opposite angles Square All the properties of a parallelogram AND rectangle AND rhombus! o Trapezoid Exactly 1 pair of opposite sides parallel The midsegment is parallel to the bases and is the average (or ½ the sum) of the bases o Isosceles Trapezoid All the properties of a trapezoid Legs are congruent, and base angles are congruent Diagonals are congruent o Kite Exactly 2 pairs of consecutive sides congruent Exactly 1 pair of opposite angles congruent (the “elbows”) Diagonals are perpendicular Prove a quadrilateral is a special quadrilateral. o On a coordinate plane, plot the 4 vertices and Use slope (rise/run) to prove parallel (same slope) or right angles (opposite reciprocals) Use Pythagorean Theorem or Distance Formula to prove congruent sides or diagonals o Parallelogram: By any of the 5 properties above, OR by showing 1 pair of opposite sides both parallel and congruent o Rectangle: It is a parallelogram AND… Congruent diagonals OR 1 right angle o Rhombus: It is a parallelogram AND… 1 pair congruent consecutive sides OR perpendicular diagonals OR 1 diagonal bisects opposite angles o Square: It is a parallelogram AND a rectangle AND a rhombus! Polygons o Interior angles of a polygon found by using (n 2)180 ⁰ o The sum of the exterior angle of a polygon = 360⁰ o A REGULAR polygon is both equilateral and equiangular o Exercises for Quadrilaterals and Polygons: Graph quadrilateral ABCD. Then determine the most precise name for each quadrilateral. Then find the perimeter of each quadrilateral. Leave your answers in simplest radical form. 7. Name __________ Perimeter __________ A(2, 3), B(-4, 3), C(-2, 6), D(1, 6) 8. Name __________ Perimeter __________ A(0, 6), B(3, 3), C(0, -5), D(-3, 3) 9. Name __________ Perimeter __________ E(0, 4), F(3, 0), G(7, 3), H(4, 7) 10. __________ QUVX is a rectangle with Q(-7, -3) and Z(-2, 1). What are the coordinates of U? 11. Parallelogram x = __________ y = __________ 12. Parallelogram x = __________ 13. Rhombus 1 = _____, 2 = _____ perimeter = __________ 14._______ Find the sum of the interior angles of a decagon. 15._______If one exterior angle of a regular polygon is 18⁰, how many sides does the polygon have? Find the values of the unknowns in each figure. 16.______ 17.______ 18._______ Chapter 8: Right Triangles and Trigonometry Apply o o o the Pythagorean Theorem and Pythagorean Triples to find unknown sides. a2 + b2 = c2 or leg2 + leg2 = hyp2 Some triples are 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41 You will often be asked to solve a multi-step problem which requires Pythagorean Theorem or triples but then asks for the perimeter, area, etc. Identify and apply special right triangle relationships to find lengths of unknown sides. 30° 1 60° 3 90° 2 OR 45° 45° 1 1 90° 2 Simplify radicals, add/subtract/multiply/divide radicals, and rationalize the denominator. Use trigonometric ratios to find unknown sides or angles of right triangles. o SOH CAH TOA: opposite sin hypotenuse adjacent cos hypotenuse opposite tan adjacent To find an unknown side, set up the equation and solve as a proportion. To find an unknown angle, press 2ND SIN or 2ND COS or 2ND TAN to get the inverse operation. Find the angle of elevation or angle of depression, or use them to find unknown sides. o Angle of elevation/depression is always starting at the HORIZONTAL and angling up or down. o An exception is if the problem asks for the angle “from the vertical” or “the angle between the ladder and the building.” o o angle between the ladder and the building Exercises for Right Triangles and Trigonometry: Do the given length describe a right triangle? 19. __________ 14, 48, 50 20. __________ 6, 7, 9 Angle of elevation Find the values of the variables. Leave your answers in simplest radical form. 21. x = __________ 22. x = __________ y = __________ 23. x = __________ y = __________ What are the measures of the two congruent angles? Find the value of x. Round lengths of segments to the nearest tenth and angle measures to the nearest degree. 24. __________ 25. __________ 26. __________ A surveyor measures the top of a building 50 ft. away from him. His angle-measuring device is 4 ft. above ground. The angle of elevation to the top of the building is 63°. How tall is the building? 27. __________ A forest ranger looking out from a ranger’s station can see a forest fire at a 35° angle of depression. The ranger’s position is 100 ft. above the ground. How far is it from the ranger’s station to the fire? 28. __________ A moving van traveled 200 mi west, then 70 mi south, then 50 mi east, and finally 100 mi north. Find the distance from the point of origin to the destination (to the nearest mile), and the angle from the origin to the destination (to the nearest degree). Find the value of each variable. Leave your answers in simplest radical form. Then find the perimeter and area of each composite figure. 29. w = __________ x = __________ y = __________ z = __________ perimeter = ____________________ 30. p = __________ q = __________ r = __________ s = __________ perimeter = ____________________ Chapter 9: Area Calculate the areas of the following shapes. o A=bh for a parallelogram or rectangle, where base and height form a right angle o A = s2 for a square o A=½bh for a triangle o A = ½(b1 + b2)h for a trapezoid o A = ½ d 1 d2 for a rhombus or a kite, where d = diagonal o A=½Pa for a regular polygon, where a = apothem, and P = perimeter o Composite shapes Find arc lengths and sector areas of circles. o o Circumference = 2r Area = r2 Exercises for Area: Use EXACT answers. 31. __________ Find the area of an equilateral triangle with side length of 6 ft. 32. __________ regular hexagon with side length of 4 cm. 33. __________ Find the area of an isosceles triangle with legs each 20 ft. long and a base 24 ft. long. 34. Area = __________ 35. Length of arc PQ = __________ 36. _______% Probability that a Area of sector PNQ = __________ random point is in shaded region Find the area of the figures below. 37. __________ 38. __________ 39. __________ 40. __________ Chapter 10 Surface Area and Volume Calculate the lateral area, total surface area, and volume of 3-D figures. o Prisms and Cylinders LA = Ph where P = perimeter of the base. SA = Ph + 2B h = distance between the two bases. V = Bh B = area of the base, which depends on the SHAPE of the base. o Pyramids and Cones LA = 12 P ℓ or LA = r ℓ for cones. o SA = V= 1 3 1 2 Pℓ+B where ℓ = slant height along a lateral face. Bh h = perpendicular height from the base to the vertex. Sphere SA = 4 r2 V = 43 r3 A hemisphere is half a sphere. o Composite figures: be careful when calculating surface area – do not include a base area if it lies inside the figure. Break it down into parts. o Oblique figures use the same formulas as right figures. Use ratios of similarity, area, and volume to solve similar solids. Exercises for Surface Area and Volume: Find the surface area (SA) and volume (V) of each figure to the nearest tenth. 41. SA = __________ V = __________ 42. SA = __________ V = ___________ 43. SA = __________44. SA = __________ V = ___________ V = __________ 45. __________ Two similar cones have heights of 9 cm and 4 cm. Find the ratio of their volumes. 46. a) __________ A cylinder a surface area of 25 cm2 , if the dimensions are double, what is the surface area of the new cylinder? b) __________ If the sphere has a volume of 50 cm3 and the dimensions are cut in half, what is the volume of the larger sphere? 47. a)___________Find the entire figure’s surface area to the nearest tenth. b) __________ Find the entire figure’s volume to the nearest tenth. 48. ______ Identify the cross section. Chapter 12 Circles Solve o o o o Write o for missing lengths or angles using: Properties of tangents Properties of chords and arcs Properties of inscribed angles and central angles Relationships between secants and tangents the standard equation for a circle in the coordinate plane Circle equation is (x – h)2 + (y – k)2 = r2 where (h, k) is the center and r is the radius Exercises for Circles: Find the measure of arc AB. 49. arc AB = ________ 50. arc AB = __________ 51. a) _____ b) _____ c) _____ d) _______ Find the value of the variable(s). Assume that lines that appear to be tangent are tangent. Round to the tenth. 52. __________ 53. __________ 56. x = ______ y= ______ 57. __________ 54. __________ 55. __________ 58. __________ 59. __________ 60. __________ Write an equation of the circle that passes through (2, 8) with center (-3, 4). Convert the following degree measure in radians. 61. __________ 90⁰ 62.__________ 120⁰ 63.__________ 330⁰ Chapter 13 Probability Geometric Probability: o A ratio that involves geometric measurements Probability o Probability of an event occurring is the ratio of the number of favorable outcomes to the number of possible outcomes o Independent events are events where the outcome of one event does NOT affect the outcome of the other events Exercises for Probability: 64.______ What is the probability of the spinner landing on C? 65.______ What is the probability of NOT landing on C? 66.______ What is the probability of the spinner landing on A or B? 67._____ What is the probability of the spinner landing on one of the first five letters of the alphabet? 67. 68._____ What is the geometric probability of a dart landing in the shaded area? 69._____ What is the geometric probability of a dart landing in the unshaded area?