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Geometry Final Exam Review – Ch. 7 SIMPLIFY each ratio completely. 40 18 1. = 2. = 24 33 Convert units to simplify each ratio. 4 days 3 ft . 6. = 7. = 2 weeks 8 inches Name:_________________________________ Hour: ____ 3. 20 = 35 4. 32 : 4 = 8. 6 yards = 4 feet 9. A basketball team won 12 games and lost 8. Reduce each ratio. 40 cm = 2 meter 5. 30 : 39 = 10. 8 cm = 6 mm 15. In the diagram, JK : KL is 7 : 2 and JL =36. Find JK and KL. 36 11. wins to losses Equation: 12. wins to the total number of games played 13. losses to wins 14. losses to the total number of games played x = _____ JK = _____ KL = ______ 16. Use the triangles to write each ratio in simplest form. XY ________ RS 17. Use the triangles to write each ratio in simplest form. AX XB = _______ = _______ XB AX ST ________ YZ AX = _______ AB BC = _______ XY XZ RT AY = _______ YC AY = _______ AC ________ Solve these proportions by cross-multiplying. Use the distributive property where needed. Show your work! 2 x2 x 4 3x 2 x 6 18. 19. 8 : 3 = x : 6 20. 21. 9 8 5 10 5 4 Proportion: Set up a PROPORTION to solve the following problems. 22. If 25 Valentine chocolate candies cost $20.00. How much will 42 Valentine chocolates cost? 23. Thomas finished 50 math problems in 20 minutes. At this rate, how many math problems can he do in 30 minutes? Proportion: Proportion: Answer = __________ Answer = __________ 24. Shapes that are SIMILAR are the same shape, but not necessarily the same _____________. 25. In SIMILAR shapes, the corresponding angles are _____________and the corresponding sides are _____________________. 1 The two polygons are similar. Write a proportion and solve for x. 26. 27. 28. Proportion to find x and solve: Proportion to find x and solve: Proportion to find x and solve: 29. 30. 31. LJK ~ _______ Scale Factor: _______ LMNP ~ _______ Scale Factor: ________ DEF ~ _______ Scale Factor: _______ Proportion to find x: Proportion to find x: Proportion to find x: Proportion to find y: Proportion to find y: Proportion to find y: 32. The two rectangles are similar. a. Find the scale factor (left to right). 33. The scale factor of two similar triangles is 4 : 7, find the ratio of the perimeters. b. Find the ratio of the perimeters of the rectangles. Determine if the triangles are similar by AA~, SSS~, SAS~, or none. Work must be shown to check proportions and/or angles! 34. 35. 36. Check Proportions and/or Angles: Check Proportions and/or Angles: Check Proportions and/or Angles: Postulate: _________ Postulate: _________ Postulate: _________ 37. 38. 39. Check Proportions and/or Angles: Check Proportions and/or Angles: Check Proportions and/or Angles: Postulate: _________ Postulate: _________ Postulate: _________ 2 Find the missing angles and set up proportions to find the missing side lengths for the similar triangles. 40. 41. Proportion to find x: Proportion to find y: Proportion to find x: Proportion to find y: mT= ______ mN = ______ mD = ______ mT = ______ mA = ______ 42. 43. "Flipped" or "Twisted" Bow Tie? "Flipped" or "Twisted" Bow Tie? PET ~ _______ Proportion to find x: CAB ~ _______ Proportion to find y: Proportion to find x: mP = ______ mATC = ______mA = ______ 44. Proportion to find y: m1 = ______ mA = ______ mE = ______ 45. Separate and label triangles here: Separate and label triangles here: EAB ~ _______ Scale Factor: ________ BAC ~ _______Scale Factor: ________ Proportion to find x: Proportion to find x: Proportion to find y: mE = ______ mD= ______ mF = ______ Proportion to find y: mD = ______ mBCA = ______mA = ______ Complete the following proportions by using the picture at the right. PQ ? PQ PR 46. ? = _______ 47. ? = _______ PS PT QS ? QR ? PQ QR 48. ? = _______ 49. ? = _______ ST PT ? ST 3 Use a proportion to solve for the missing length. 49. Proportion to find x: 50. Proportion to find x: 51. 52. Proportion to find z: Proportion to find x: Separate the picture into two labeled triangles and find the missing information. 53. Separate the picture into two labeled triangles. 54. Separate the picture into two labeled triangles Proportion to find x: Proportion to find y: Proportion to find x: Proportion to find y. Use the MIDSEGMENT FORMULA to solve for the length of the variable. 55. 56. 57. 58. x = ________ y = ________ a = ________ b = ________ 4 Geometry Final Exam Review – Ch. 8 Name:_________________________________ Hour: ____ 1. How do you find the perimeter of any shape?__________________________ Find the perimeter of each shape. 2. 3. 4. 5. 6-7. Draw a rectangle with the following dimensions. 6. Draw a rectangle with perimeter 12 and area 8. 7. Draw a rectangle with perimeter 10 and area 4. Converting units. Fill in the blanks. 8. 1 yd = _____ feet 9. 2 yd = _____ feet 10. 6 feet = ____ yards 11. 12 feet = ____yards 12. 1 foot = ___ inches 13. 5 feet = ___inches 14. 24 inches = ___feet 15. 48 inches = ____ft 16. 1 meter = ____cm 17. 4 meters = ____cm 18. 5 cm = ____mm 19. 12 cm = ______mm 20. How do you find the area of a rectangle?_________________ 21. How do you find the area of a parallelogram?_______________ Find the area of each rectangle or parallelogram. 22. 23. 24. 26. 27. 28. 25. 29. Given the dimensions of a parallelogram, find the area. 30. base = 24 cm 31. base = 18 in 32. base = 16.2 m height = 5 cm height = 25 in height = 9.4 m 5 33. base = 45 ft height = 8 yd Find x for each parallelogram. 34. Area = 48 cm2 35. Area = 63 in2 36. How do you find the area of a triangle?_____________________ Find the area of each triangle. 37. 38. 39. 40. Find x for each triangle. 41. Area = 32 in2 42. Area = 24 in2 Fill in the following formulas. 43. Area of a trapezoid _____________________ 44. Area of a rhombus __________________________ 45. Area of a regular polygon ________________ Find the area of each shape. 46. A trapezoid with bases of length 10in and 12 in, and height 7 in. 47. A rhombus with diagonals of length 14in and 6 in. 48. A regular octagon with sides of length 8 mm and apothem of 9.7 mm. 49. A regular pentagon with sides length 20 cm and apothem 13.7 cm. 50. 51. 52. Find the area of the shaded region. 6 53. Find the area of the shaded region. 54. 55. 56. Area rectangle=_______ Area triangle=_______ Shaded area=_______ Area rectangle=_______ Area triangle=_______ Shaded area=_______ 57. Area parallelogram=_______ Area rectangle=_______ Shaded area=_______ Match the name of each polygon with the number of sides. 57. Octagon_____ 61. Nonagon _____ 58. Hexagon _____ 62. Pentagon _____ 59. Heptagon _____ 63. Decagon _____ 60. Quadrilateral _____ 64. Triangle _____ Classify (Name) the polygon by its number of sides. 65. 66. 67. Area parallelogram=_______ Area square=_______ Shaded area=_______ A. 3 sides B. 4 sides C. 5 sides D. 6 sides 68. E. 7 sides F. 8 sides G. 9 sides H. 10 sides 69. 70. How many DIAGONALS from point X does each polygon have? 71. A REGULAR POLYGON has all equal ____________ and all equal __________ Draw and label a regular polygon with… 3 sides 4 sides 5 sides 6 sides 8 sides 72. Fill in the chart. Name Picture # of Sides No picture n SUM of Interior ∠’s EACH interior Angle Triangle Quadrilateral Pentagon Hexagon FORMULAS Memorize them! 7 SUM of Exterior ∠’s EACH Exterior Angle For each regular polygon, find the SUM of the interior angles and the measure of EACH interior angle. 73. Octagon (8 sides) 74. Polygon with 15 sides 75. Polygon with 20 sides SUM of Interior Angles ____________ SUM of Interior Angles ____________ SUM of Interior Angles ____________ EACH interior angle ____________ EACH interior angle ____________ EACH interior angle ____________ Find the measure of the missing angle. 76. 77. 78. 78. # of sides___ Sum of Interior ∠’s _____ # of sides___ Sum of Interior ∠’s ____ # of sides___ Sum of Interior ∠’s ____ m∠D =_______ m∠A =_______ m∠D =_______ 79.79. # of sides___ Sum of Interior ∠’s ____ m∠A =_______ Given the SUM of the INTERIOR angles, work backwards to find the number of SIDES in each shape. 80. 720° 81. 1080° 82. 1620° 83. 2880° For each regular polygon, find the SUM of exterior angles and the measure of EACH exterior angle. 84. Octagon (8 sides) 85. Polygon with 15 sides 86. Polygon with 20 sides SUM of Exterior Angles ____________ SUM of Exterior Angles ____________ SUM of Exterior Angles ____________ EACH Exterior angle ____________ EACH Exterior angle ____________ EACH Exterior angle ____________ Write an equation and solve for x. 87. 88. 89. SUM of Exterior Angles ____________ Equation: SUM of Exterior Angles ____________ Equation: SUM of Exterior Angles ____________ Equation: 90. SUM of Exterior Angles ____________ Equation: Given the measure of EACH EXTERIOR angle in a regular polygon, work backwards to find the number of SIDES. 91. 12° 92. 120° 93. 90° 94. 45° Classify each figure as CONVEX (“caved out”) polygon, CONCAVE (“caved in”) polygon or Not a Polygon. 95. 96. 97. 98. 8 Ch. 8 CIRCLE Final Exam Review MATCH the key word with the descriptive phrase. ____1. The set of all point in a plane that are the same distance from a given point ____2. The distance from the center to a point on the circle ____ 3. The distance across the circle, through the center ____4. The distance around a circle ____ 5. The amount of surface covered by a circle ____6. A portion of the AREA of a circle ____7. A portion of the CIRCUMFERENCE of a circle A. diameter B. radius C. circle D. circumference E. area F. arc length G. area of sector 8. What is the relationship between the radius and the diameter?_______________ 9. If the radius is 7 cm, then the diameter is _______ If the diameter is 18 m, then the radius is __________ 10. State the FORMULA for CIRCUMFERENCE of a circle: ________________ – NOT 3.14 APPROX: use 3.14 – Find the exact and approximate CIRCUMFERENCE of each circle. Your answers should have plain units. 11. 12. 13. Diameter = 20 mm 14. Radius = 4 cm EXACT circumference __________ EXACT circumference __________ EXACT circumference __________ APPROX circumference __________ APPROX circumference __________ APPROX circumference __________ 15. A farmer wants to build a circular pen for his chicken. He wants the radius of his pen to be 25 ft. Approximately how many feet of fencing would he need to build the pen? EXACT circumference __________ APPROX circumference __________ 16. A bicycle tire has a diameter of 24 inches. How far does the tire go in ONE revolution? (Hint: use circumference formula) Which formula will you use for fencing? Circumference or Area? How far does the tire go in 10 revolutions? Given the CIRCUMFERENCE, work backwards using the formula to find the diameter and radius. 17. Circum= 50 19. Circum = 18.84 20. Circum = 21.98 d = ________ r = _________ d = ________ r = _________ d = ________ r = _________ d = ________ r = _________ 21. State the FORMULA for finding ARC LENGTH: ___________________ Use the formula above to find the arc length of AB. 22. 23. 24. 9 25. 26. State the FORMULA for AREA of a circle: ________________ – NOT 3.14 APPROX: use 3.14 – Find the exact and approximate AREA of each circle. Your answers should have square units. 27. 28. 29. Radius = 3 m 30. Diameter = 8 in. EXACT area __________ EXACT area __________ EXACT area __________ EXACT area __________ APPROX area __________ APPROX area __________ APPROX area __________ APPROX area __________ Given the AREA, work backwards using the formula to find the radius. 2 31. Area = 36 32. Area = m2 33. Area = 78.5 ft2 34. Area = 12.56 cm2 35. State the FORMULA for finding the AREA OF A SECTOR: _____________________ Find the area of each sector. 36. 37. 38. 39. Find the area of the shaded region. 40. 41. 42. 43. Exact area of big circle:_________ Exact area of square:_________ Exact area of rectangle:________ Exact area of parallelogram:_____ Exact area of small circle:_______ Exact area of circle:_______ Exact area of circle:_______ Area of Shaded region: _________ Area of Shaded region: ______ Area of Shaded region: _______ Exact area of circle:_______ Area of Shaded region: _________ A pizza is cut into 8 congruent pieces as shown. The diameter of the pizza is 16 inches. 44. Find the circumference of the pizza. 46. Find the radius of the pizza. 47. Find the area of the top of the entire pizza. 10 Geometry Final Exam Review – Ch. 9 Name:_________________________________ Hour: ____ Tell whether the solid is a polyhedron. If so, name the solid. 1. 2. 3. 4. Name the polyhedron. Then count the number of faces and edges. 5. 6. 7. Name: Name: Name: Faces: Faces: Faces: Edges: Edges: Edges: Use Euler’s formula F + V = E + 2 to find the number of faces, edges or vertices. 8. A prism has 4 faces and 6 edges. How many vertices does it have? 9. A pyramid has 5 faces and 6 vertices. How many edges does it have? 10. A pyramid has 12 edges and 7 vertices. How many faces does it have? 11 Name the solid, then find the surface area to the nearest whole number. 11. 12. 13. Name: Name: Name: 14. 15. 16. Name: Name: Name: 17. 18. 19. Name: Name: Name: 20. 21. 22. Name: Name: Name: 12 Name the solid. Then find the volume of the solid. 23. 24. 25. Name: Name: Name: 26. 27. 28. Name: Name: Name: 29. 30. 31. Name: Name: Name: 32. 33. 34. Name: Name: Name: 13 Geometry Final Exam Review – Ch. 10 Find the value of each expression. 1. = ____ 2. = ______ Name:_________________________________ Hour: ____ 3. 122 = _______ 4. 82 = _____ List the perfect squares from 1 to 225 Use the list of perfect squares to simplify each radical…show EXACT answers only. NO DECIMALS! 5. 6. 7. 8. 9. 10. 11. 12. Use the calculator to find the following rounded to the nearest 100th (two decimal places). 13. _________ 14. __________ 15. _________ 16. __________ 17. State the Pythagorean Theorem: ______________________________ What is it used for?______________________ Can the given side lengths make a right triangle. Answer Yes or No. YOU MUST SHOW WORK! 18. 12, 23, 35 19. 5, 13, 12 20. Use the Pythagorean Theorem to find the following missing sides. An equation MUST be given. Round to 2 decimals, if necessary. 21. 22. 23. Equation: ________________ x = _______ Equation: ________________ x = _______ Equation: ________________ x = _______ 24. 25. 26. Equation: ________________ x = _______ Equation: ________________ x = _______ Equation: ________________ x = _______ 14 27. A 15-foot ladder is leaning against a wall. It reaches up the wall 10 feet. How far is the bottom of the ladder from the wall? 28. A 30-ft wire is attached to an electrical pole. The wire attaches to a stake on the ground. If the stake is 18 feet from the base of the pole, How tall is the pole? Equation: Equation: 29. How long is the hypotenuse of a doorway that is 9 feet by 4 feet? 30. A helicopter flies 9 miles due east and then 6 miles due south. How far is if from its starting point? Equation: Equation: Can a mattress that is 10 feet long fit through the doorway?________ Remind yourself of the 45-45-90 and 30-60-90 triangle rules! 45-45-90: hypotenuse = leg 30-60-90: hypotenuse = short leg Use the special triangle rules to find the missing sides of the following triangles. 31. 32. 33. x = _______ x = _______ y = ______ 34. x = _______ x = _______ y = ______ 35. y = ______ y = ______ 36. x = _______ y = ______ x = _______ y = ______ 37. Us a CALCULATOR set in DEGREE mode to find the following values. Round answers to nearest hundredth. a) Sin 45 = ________ b) tan 30 = _______ c) cos 90 = ______ d) cos 60 = ______ e) sin 60 = ________ Fill in the ratios for each trig function using the words: opposite, adjacent and hypotenuse. How do we remember these definitions? _________________________________________ 15 For each triangle, give the sin, cos, and tan in fraction form. Find the missing sides where needed and reduce all fractions! 38. 39. 40. a = ____ sin A_____ sin B_______ sin A_____ sin B_______ sin A_____ sin B_______ cos A______ cos B_______ cos A______ cos B_______ cos A______ cos B_______ tan A ______ tan B ______ tan A ______ tan B ______ tan A ______ tan B ______ Use sin, cos, or tan proportion to solve for the variable. 41. 42. a = ________ 43. 43. a = ________ 44. Donovan leans a 15-ft ladder against the wall. The ladder makes a 70° angle with the ground. How far up the building does the ladder reach? a = ________ 45. A tree casts a shadow 25 feet long when the angle of elevation to the sun is 68°. How tall is the tree? Use SOH CAH TOA to find the missing ANGLE. Write an equation and use the INVERSE to find the angle measure. Round to nearest 100th of a degree. 46. 47. m∠A = ____________ m∠A = ___________ 48. Stefan leans a 20-ft ladder against a wall. The base of the ladder is 3 feet from the wall. What ANGLE does the ladder make with the ground? 49. Chelsea visited the Washington Monument which is 550ft tall on her summer vacation. She stood 400 feet away from the base of the monument to take a picture. At what ANGLE did she need look up to ensure that she captured the top of the monument in her picture? 16 Geometry Final Exam Review – Ch. 11 Name: ________________________ Hour: _____ 1. How many degrees are in a circle?___________ 2. How many degrees are in a semicircle? _____________ Name each of the following for circle O. 3. A semicircle __________ 4. Two minor arcs __________ and __________ 5. Two major arcs__________ and __________ 6. In a circle, the measure of the central angle is the ____________ the measure of the arc. Find the measure of each angle for each arc of circle P. 7. m∠SPR______________ 8. ________________ 9. 11 . ________________ ________________ 10. ________________ 12. ________________ 13. In a circle, the measure of the inscribed angle is the _____________ the measure of the arc. 14. What is the measure of an angle that is inscribed in a semicircle?________________ Find the measure of the following angles and arcs. 15. 16. 17. 18. What is a tangent segment?_________________ 19. What kind of angle is formed when a radius and a tangent meet? ________________ 20. If two tangent segments are drawn from a point outside the circle, these segments are ___________ Find the lengths of the following segments. 21. 22. 23. SR = _____ OT = ______ OC = ____ OB = _____ AB = ____ MT = _________ 24. Equal chords mean _________ arcs. 25. If a diameter is perpendicular to a chord, then it_______ the chord and the arc. 17 Using the given picture, find the following lengths. 26. PB = _____________ 27. PC = ______________ 28. PE = _____________ 29. CE = ______________ Note: PD = 5, BE = 2 30. AE = ____________ Draw the following. 31. a triangle inscribed in a square 32. A circle inscribed in a triangle 33. A triangle circumscribed about a circle What is the rule for finding the angle in a picture that is Chord-Chord Find the following angles. 34. m∠1=___________ 35. m∠1=_______ m∠2=_________ ____________________________ 36. m∠1=_______ m∠2=_________ What is the rule for finding the angle in a picture that is tangent-chord Find the following angles. 37. m∠1=___________ 38. m =_______ m∠1=_________ _______________________ 39. m∠1=_______ m∠2=_________ What is the rule for finding angle 1 in each of the following pictures ___________________ Find the following angles. 40. m∠1=___________ 41. m∠1=_________ 42. m Write an equation and solve for x. 43. 44. 45. Equation:________________ Equation:________________ Equation:________________ Answer: _________________ Answer: _________________ Answer: _________________ 18 =_______ m∠1=_________