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GMAT Geometry - everything you need to know

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This slideshow features 194 screenshots from GMAT Prep Now’s entire Geometry module (consisting of 42 videos). It covers every key concept you need to know about GMAT Geometry. It also includes 27 practice questions.

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GMAT Geometry - everything you need to know

  1. 1. GMAT Geometry - Everything you need to know This slideshow features screenshots from GMAT Prep Now’s entire Geometry module (consisting of 42 videos). It covers every key concept you need to know about GMAT Geometry. It also includes 27 practice questions. www.GMATPrepNow.com
  2. 2. GMAT Geometry - Everything you need to know www.GMATPrepNow.com Note: since these slides are just snippets of a full-length video course, there may be times when you’re unable to glean all the relevant information from a particular screenshot. If, at any time, you’d like to watch the entire video on a certain topic, just click on the link at the top of that page, and you’ll be taken that that particular video.
  3. 3. GMAT Geometry - Everything you need to know If you enjoy this unique learning format, let us know, and we’ll add similar resources to our SlideShare page
  4. 4. Lines and Angles (watch the entire video here)
  5. 5. Lines and Angles l line: a straight path that extends without end in both directions (watch the entire video here)
  6. 6. Lines and Angles l   A B AB: line segment AB: length of line segment AB (e.g., DE=7) line: a straight path that extends without end in both directions (watch the entire video here)
  7. 7. Lines and Angles   55 A   B C 55 55 ABC CBA      55x  x angle: intersection of 2 lines : measured in degrees or radians (watch the entire video here)
  8. 8. Lines and Angles  180 Angles on a line add to 180° a cb 180a b c   70x 70 180 110 x x    (watch the entire video here)
  9. 9. Lines and Angles 90 right angle: angle of 90 degrees  P PQ is perpendicular to AB  BA  Q (watch the entire video here)
  10. 10. Lines and Angles bisect: cut or divide into 2 equal pieces J JK bisects AB  BA   A   B C bisects ABC bisectoris the of ABC line l is the perpendicular bisector of AB  BA K  l (watch the entire video here)
  11. 11. Lines and Angles a c x x b d - a and c are vertical angles - a and c are opposite angles - a and c are vertically opposite angles - b and d are opposite angles Opposite angles are equal y y Aside: 180x y   (watch the entire video here)
  12. 12. Lines and Angles w 50 yx (watch the entire video here)
  13. 13. Lines and Angles w 50 yx 50x  50 180 130 w w    130y  Opposite angles are equal Angles on a line add to 180° (watch the entire video here)
  14. 14. Lines and Angles 1 2 If two lines do not intersect, they are parallel 1 2 (watch the entire video here)
  15. 15. Lines and Angles 1 2 If two lines do not intersect, they are parallel y y y y x Note: 180x y  x x x 1 2 (watch the entire video here)
  16. 16. Lines and Angles Opposite angles are equal Angles on a line add to 180° 1 2 1 2 y y y y x x x x (watch the entire video here)
  17. 17. Practice Question A) 10 B) 17.5 C) 22 D) 35 E) 42.5 If l1 and l2 are parallel, then x = 1 2  3 5x   15x  Note: Figure not drawn to scale
  18. 18. A) 10 B) 17.5 C) 22 D) 35 E) 42.5 If l1 and l2 are parallel, then x = 1 2  3 5x   15x   3 5x     15 3 5 180 4 10 180 4 170 42.5 x x x x x                 Note: Figure not drawn to scale Practice Question (watch the entire video here)
  19. 19. Triangles – Part I (watch the entire video here)
  20. 20. Triangles – Part I A B C w x y 180w x y   Angles in a triangle add to 180° (watch the entire video here)
  21. 21. Triangles – Part I A B C 21 44 180w x y   Angles in a triangle add to 180° w (watch the entire video here)
  22. 22. Triangles – Part I A B C 21 44 180w x y   Angles in a triangle add to 180° w 180 180 1 2 4 5 4 1 1 65 w w w       (watch the entire video here)
  23. 23. Triangles – Part I A B C w x y The longest side is opposite the largest angle The shortest side is opposite the smallest angle A B C a b c If thena b c A B C    (watch the entire video here)
  24. 24. Triangles – Part I 1 The sum of the lengths of any two sides of a triangle must be greater than the third side. 2 4 1 2 1 42  4 (watch the entire video here)
  25. 25. Triangles – Part I If a triangle has sides with lengths 3 and 7, what lengths are possible for the third side? 3 7 The sum of the lengths of any two sides of a triangle must be greater than the third side. (watch the entire video here)
  26. 26. Triangles – Part I If a triangle has sides with lengths 3 and 7, what lengths are possible for the third side? 7 third side 73 37     3 4 rd difference between other 2 sides 3 side sum of other 2 sides  Given lengths of sides A and B rd 3 sideA B A B    (watch the entire video here)
  27. 27. Triangles – Part I Given lengths of sides A and B rd 3 sideA B A B    Angles in a triangle add to 180° A B C a b c If thena b c A B C    The sum of the lengths of any two sides of a triangle must be greater than the third side. (watch the entire video here)
  28. 28. Is w > x? Q P w x y R 2) 3QR  1) 6PQ  Practice Question A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient D) EACH statement ALONE is sufficient E) Statements (1) and (2) TOGETHER are NOT sufficient
  29. 29. Q P w x y R 1) 6PQ  A B C a b c If thena b c A B C    2) 3QR  3 1&2) 6 Given lengths of sides A and B rd 3 sideA B A B    3 9PR  E 6 3 36PR    Is w > x? Is ?PR PQ Practice Question (watch the entire video here) INSUFFICIENT INSUFFICIENT INSUFFICIENT
  30. 30. What is the value of x in terms of y ? A) 65 B) 21 C) 22 D) 21 E) 22 y y y y y      x y 22 43 Practice Question
  31. 31. (watch the entire video here)Practice Question What is the value of x in terms of y ? A) 65 B) 21 C) 22 D) 21 E) 22 y y y y y      x y 22 43 a 43 180ya    43 22 180xa        43 22 43 43 22 43 22 22 a x y x y x y x y a              Angles in a triangle add to 180° Solution #1
  32. 32. Solution #2 (watch the entire video here)Practice Question What is the value of x in terms of y ? A) 65 B) 21 C) 22 D) 21 E) 22 y y y y y      x y 22 43  158 x   180 158 18 1 0 22 22 22 58y y x y x y x y x x            Angles on a line add to 180° 1 22 180 1 58 58 x x c c c x        158 180 22 y x y x     
  33. 33. Assumptions and Estimation (watch the entire video here)
  34. 34. Assumptions and Estimation 120  • Lines that appear straight can be assumed to be straight (watch the entire video here)
  35. 35. Assumptions and Estimation 120  60 • Lines that appear straight can be assumed to be straight (watch the entire video here)
  36. 36.  • Do not make assumptions about angle measurements x Assumptions and Estimation (watch the entire video here)
  37. 37. y  • y+x =180 • Both angles are greater than zero degrees x Assumptions and Estimation   (watch the entire video here)
  38. 38. • Do not make assumptions about parallelism 1 2 1 2 Assumptions and Estimation (watch the entire video here)
  39. 39. Problem Solving Questions • Figures are drawn to scale unless stated otherwise • Estimate to confirm calculations and guide guesses  x 40 O BE A) 40 B) 50 C) 60 D) 70 E) 80 Assumptions and Estimation C DA If is the center of the circle, and , what is the value of ? O AB CD x (watch the entire video here)
  40. 40. Data Sufficiency Questions • Figure conforms to information in question • Figure does not necessarily conform to information in statements • Avoid visual estimation Assumptions and Estimation (watch the entire video here)
  41. 41. Assumptions and Estimation • Lines that appear straight can be assumed to be straight • Angles are greater than zero degrees • Do not make assumptions about angle measurements • Do not make assumptions about parallelism • Use visual estimation sparingly (watch the entire video here)
  42. 42. Geometry Strategies – Part I (watch the entire video here)
  43. 43. Geometry Strategies – Part I • Redraw figures • Add all given information • Add all information that can be deduced • Add/extend lines • Assign variables and use algebra • • Drawn to scale  estimate to confirm calculations and guide guesses • Drawn to scale  estimate measurements to confirm or guess (watch the entire video here)
  44. 44. Triangles – Part II (watch the entire video here)
  45. 45. Triangles – Part II Isosceles triangle • 2 equal sides, 2 equal angles A B C a b c If thena b c A B C    40 40 100 x x (watch the entire video here)
  46. 46. Triangles – Part II 38 (watch the entire video here)
  47. 47. Triangles – Part II 38 38 104 (watch the entire video here)
  48. 48. Triangles – Part II 38 38 104 40 (watch the entire video here)
  49. 49. Triangles – Part II 38 38 104 40 x x 40 180 2 40 180 2 140 70 x x x x x        (watch the entire video here)
  50. 50. Triangles – Part II 38 38 104 40 70 40 180 2 40 180 2 140 70 x x x x x        70 (watch the entire video here)
  51. 51. Triangles – Part II A B C Equilateral triangle • 3 equal sides, 3 equal angles 60 60 60 (watch the entire video here)
  52. 52. Triangles – Part II A B C 10 4 8 Area - ft2 - cm2 - m2 (watch the entire video here)
  53. 53. Triangles – Part II A B C base height Area 2   10 4 8 Area 1 Area base height 2   (watch the entire video here)
  54. 54. Triangles – Part II A B C base height Area 2   10 Area 15 3 2    10 3 4 8 altitude height Area (watch the entire video here)
  55. 55. Triangles – Part II 10 4 8 A B C 7.5 base height Area 2   7 A . re 5 a 15 4 2    Area (watch the entire video here)
  56. 56. Triangles – Part II A B C 60 60 60   2 3 side Area 4   (watch the entire video here)
  57. 57. Triangles – Part II A B C 60 60 60   2 3 side Area 4   6 6 6   2 3 Area 4 3 36 4 9 3 6     (watch the entire video here)
  58. 58. Triangles – Part II 60 60 60 The altitudes of isosceles triangles and equilateral triangles bisect the base. (watch the entire video here)
  59. 59. Triangles – Part II • An isosceles triangle has 2 equal sides and 2 equal angles • An equilateral triangle has 3 equal sides and 3 equal angles (60° each) base height Area 2     2 3 side Area 4   • The altitudes of isosceles triangles and equilateral triangles bisect the base (watch the entire video here)
  60. 60. Practice Question A) 27.5 B) 55 C) 62.5 D) 70 E) 125 If AB and CD are parallel, and AB= BC, then x = A B C D x 55 Note: Figure not drawn to scale
  61. 61. Practice Question A) 27.5 B) 55 C) 62.5 D) 70 E) 125 If AB and CD are parallel, and AB= BC, then x = Note: Figure not drawn to scale A B C D x 5555         55 180 110 18 5 5 70 5 5 0 x x x       (watch the entire video here)
  62. 62. Right Triangles (watch the entire video here)
  63. 63. Right Triangles leg1 • Right triangle: triangle with right (90°) angle • The hypotenuse is the longest side leg2       2 2 2 1 2leg leg hypotenuse  2 2 2 a b c  a b c 2 2 2 a b c  a bc 2 2 2 a b c  a bc (watch the entire video here)
  64. 64. Right Triangles 8 6 x (watch the entire video here)
  65. 65. Right Triangles 2 2 2 a b c  a bc 8 6 x 2 2 2 2 2 8 6 64 36 100 100 10 x x x x x        (watch the entire video here)
  66. 66. Right Triangles 2 2 2 a b c  a bc 8 6 x 2 2 2 2 2 8 6 64 36 100 100 10 x x x x x        6 4 x (watch the entire video here)
  67. 67. Right Triangles 2 2 2 a b c  a bc 8 6 x 2 2 2 2 2 8 6 64 36 100 100 10 x x x x x        2 2 2 a b c  a bc 6 4 x 2 2 2 2 2 4 6 16 36 20 20 2 5 x x x x x        4 5 2 5 x x   (watch the entire video here)
  68. 68. Right Triangles • 3-4-5 4 35 • 5-12-13 12 13 5 • 8-15-17 2 2 2 3 4 5  2 2 2 5 12 13  • 7-24-25 Pythagorean triples: A set of 3 integers that can be the sides of a right triangle (watch the entire video here)
  69. 69. Right Triangles 8x 17 15 • 8-15-17 2 2 2 15 17x   2 2 2 a b c  (watch the entire video here)
  70. 70. Right Triangles • 3-4-5 • 5-12-13 • 8-15-17 • 7-24-25 6-8-10 9-12-15 12-16-20 10-24-26 4 35   4 7 28   5 7 35    213 7x   . . . . . . . . . 2 corresponding sides required to use Pythagorean triples . . . (watch the entire video here)
  71. 71. Right Triangles • 3-4-5 • 5-12-13 • 8-15-17 • 7-24-25 6-8-10 9-12-15 12-16-20 10-24-26 . . . . . . . . . 50 4 35 Enlarged by factor of 10 50 24 7 25 Enlarged by factor of 2 40 30 48 14 2 corresponding sides required to use Pythagorean triples . . . (watch the entire video here)
  72. 72. Right Triangles • 3-4-5 • 5-12-13 • 8-15-17 • 7-24-25 6-8-10 9-12-15 12-16-20 10-24-26 . . . . . . . . . 3 4 x . . . (watch the entire video here)
  73. 73. Right Triangles • 3-4-5 • 5-12-13 • 8-15-17 • 7-24-25 6-8-10 9-12-15 12-16-20 10-24-26 . . . . . . . . . 3 x 4 2 2 2 a b c  2 2 2 2 2 3 4 9 16 7 7 x x x x       . . . (watch the entire video here)
  74. 74. Right Triangles 2 2 2 a b c  a bc • Watch out for Pythagorean triples (and their multiples) 3-4-5 5-12-13 8-15-17 7-24-25 (watch the entire video here)
  75. 75. Practice Question A A) 2 3 B) 2 5 C) 30 D) 4 3 E) 4 5 B  The height of this rectangle is twice its width. If the distance between points A and B is , what is the rectangle’s height?60
  76. 76. Practice Question A A) 2 3 B) 2 5 C) 30 D) 4 3 E) 4 5 x 2x        22 2 2 2 2 2 2 60 4 60 5 60 12 12 4 3 2 3 x x x x x x x x x         B  60 2 2 2 a b c   2 4 3 2 2 3x   The height of this rectangle is twice its width. If the distance between points A and B is , what is the rectangle’s height?60 (watch the entire video here)
  77. 77. Practice Question A) 21 B) 9 C) 2 21 D) 149 E) 3 21 If the rectangular box shown here is 6 inches wide, 8 inches long and 7 inches high, what is the distance, in inches, between points A and B ? B A   8 6 7
  78. 78. A) 21 B) 9 C) 2 21 D) 149 E) 3 21 B A   8 6 7 If the rectangular box shown here is 6 inches wide, 8 inches long and 7 inches high, what is the distance, in inches, between points A and B ? 10 x 7 A B 10 x 2 2 2 a b c  2 2 2 2 2 10 7 100 49 149 149 x x x x       Practice Question (watch the entire video here) Solution #1
  79. 79. Practice Question A) 21 B) 9 C) 2 21 D) 149 E) 3 21 If the rectangular box shown here is 6 inches wide, 8 inches long and 7 inches high, what is the distance, in inches, between points A and B ? A B w x y 2 2 2 AB w x y   2 2 2 8 6 7 64 36 49 149 AB        B A   8 6 7 (watch the entire video here) Solution #2
  80. 80. Special Right Triangles 45-45-90 triangle 1 45 2 2 1.4 45 1 leg : leg : hypotenuse 1 : : x : : 1 x 2x 2 30-60-90 triangle 1 30 60 2 3 3 1.7 3 leg : leg : hypotenuse 1 : : 2 3xx : : 2x (watch the entire video here)
  81. 81. Special Right Triangles 12 30 x y (watch the entire video here)
  82. 82. Special Right Triangles 12 30 x y 30 60 1 2 3 60 enlargement factor: 6 (watch the entire video here)
  83. 83. Special Right Triangles 12 30 x y 30 60 1 2 3 60 enlargement factor: 6  61 6 x    3 6 3 6y   (watch the entire video here)
  84. 84. Special Right Triangles 5 2 x 5 2 (watch the entire video here)
  85. 85. Special Right Triangles 5 2 x 5 2 45 1 2 45 1 45 45 enlargement factor:     2 5 4 5 2 2 10 5x     5 2 (watch the entire video here)
  86. 86. Special Right Triangles 45 45 60 60 30 Square Equilateral Triangle Watch out for special right triangles “hiding” in squares and equilateral triangles (watch the entire video here)
  87. 87. Special Right Triangles 45 1 2 45 1 30 60 1 2 3 (watch the entire video here)
  88. 88. Practice Question A) 3 2 B) 2 6 C) 4 3 D) 6 2 E) 6 3 B  A C D If , 6 and 105 , thenAD BD AB ABC x     Note: Figure not drawn to scale x
  89. 89. Practice Question A) 3 2 B) 2 6 C) 4 3 D) 6 2 E) 6 3 B  A C D If , 6 and 105 , thenAD BD AB ABC x     Note: Figure not drawn to scale 45 45 60 30 x 45 1 45 1 enlargement factor: ? 6 2 26 6 2 30 60 2 3 6 2 1 2 12 2 2 2 12 2 2 2 6 2 6 x            (watch the entire video here)
  90. 90. Similar Triangles (watch the entire video here)
  91. 91. Similar Triangles Similar triangles have the same 3 angles in common 40 20 120 40 20 120 With similar triangles, the ratio of any pair of corresponding sides is the same w a b c x y a w b c x y   (watch the entire video here)
  92. 92. Similar Triangles  * * x 5 7 9 6 (watch the entire video here)
  93. 93. Similar Triangles  * *   x 5 7 9 With similar triangles, the ratio of any pair of corresponding sides is the same    5 5 63 6 5 3 5 7 9 7 9 x x x x     6 (watch the entire video here)
  94. 94. Similar Triangles Similar triangles have the same 3 angles in common 40 20 120 40 20 120 With similar triangles, the ratio of any pair of corresponding sides is the same w a b c x y a w b c x y   (watch the entire video here)
  95. 95. Practice Question If , thenABC BCD x    Note: Figure not drawn to scale BA C D 8 10 12 5 x E A) 4 25 B) 6 C) 6 36 D) 5 E) 24
  96. 96. Practice Question If , thenABC BCD x    Note: Figure not drawn to scale BA C D x   E   ❤ ❤ With similar triangles, the ratio of any pair of corresponding sides is the same    12 5 5 12 10 12 1 50 50 12 25 6 0 x x x x x      A) 4 25 B) 6 C) 6 36 D) 5 E) 24 8 10 12 5 (watch the entire video here)
  97. 97. Quadrilaterals (watch the entire video here)
  98. 98. Quadrilaterals Angles in a quadrilateral add to 360° A D C w x y 360w x y z    B z (watch the entire video here)
  99. 99. Quadrilaterals square rectangle trapezoid parallelogram rhombus (watch the entire video here)
  100. 100. Quadrilaterals parallelogram     opposite sides parallel rectangle opposite sides parallel all angles are 90   rhombus  opposite sides parallel all sides are equal     square opposite sides parallel (watch the entire video here)
  101. 101. Quadrilaterals trapezoid 2 sides parallel (watch the entire video here)
  102. 102. Quadrilaterals Rhombus (and square) • diagonals are perpendicular bisectors Rectangle (and square) • diagonals are equal length A D C B AC BD (watch the entire video here)
  103. 103. Quadrilaterals square rectangle trapezoid area base height  base base height height base2 base1 height 1 2base base area height 2 average of bases height          parallelogram rhombus base height base height (watch the entire video here)
  104. 104. Quadrilaterals rhombus 1 2diagonal diagonal area 2  4 7 area 2 28 2 14 4 7    (watch the entire video here)
  105. 105. Quadrilaterals Angles in a quadrilateral add to 360° parallelogram     opposite sides parallel rectangle opposite sides parallel all angles are 90   rhombus  opposite sides parallel all sides are equal     square opposite sides parallel trapezoid 2 sides parallel area base height  (watch the entire video here)
  106. 106. Polygons (watch the entire video here)
  107. 107. Polygons Polygon: Closed figure formed by 3 or more line segments     (watch the entire video here)
  108. 108. Polygons “polygon” “convex polygon” (all interior angles less than 180°)   (watch the entire video here)
  109. 109. Polygons b a 180a b c   Triangle Quadrilateral Pentagon c b a c d 360a b c d    b a c d 540a b c d e     e Hexagon b a c d 720a b c d e f      ef (watch the entire video here)
  110. 110. Polygons        The sum of the interior angles in an N-sided polygon is equal to  180 2N  6 1 2 3 4 5 Octagon       sum of angles 180 2 8180 2 180 10 6 80 N      (watch the entire video here)
  111. 111. Polygons Regular polygon: equal sides and equal angles regular pentagon     (watch the entire video here)
  112. 112. Polygons • Polygon: Closed figure formed by 3 or more line segments • “polygon” “convex polygon” (all interior angles less than 180°) Triangle Quadrilateral Pentagon Hexagon • Regular polygon: equal sides and equal angles The sum of the interior angles in an N-sided polygon is equal to  180 2N  (watch the entire video here)
  113. 113. Circles (watch the entire video here)
  114. 114. Circles Circle: set of points that are equidistant from a given point  center  A B  C  E D diameter u2 radi s  arc - “arc CDE ”  - “minor arc CE”   (watch the entire video here)
  115. 115. Circles    circumference 2 radius 2 r       Circumference 3.14 3 22 7     circumference diameter d        (watch the entire video here)
  116. 116. Circles   circumference 2 r Circumference     circumference 2 16 feet 16 3 48 f 8 eet       8 ft (watch the entire video here)
  117. 117. Circles Area   2 area r   2 2 area 6 8 4 ft     8 ft (watch the entire video here)
  118. 118. Circles  center  A B  C  E  circumference diameter   circumference 2 radius   3.14 3    2 area r arc   (watch the entire video here)
  119. 119. Practice Question A) 9 B) 12 C) 15 D) 18 E) 36      If is the center, 45 , and 6,then the area of the circle isO OBC BC    C B O Note: Figure not drawn to scale
  120. 120. Practice Question A) 9 B) 12 C) 15 D) 18 E) 36      If is the center, 45 , and 6,then the area of the circle isO OBC BC   CO Note: Figure not drawn to scale 45  4590 B With similar triangles, the ratio of any pair of corresponding sides is the same 6 2 area r 2 area 36 2 18 6 2                   r 12 6 2 6 r r   (watch the entire video here)
  121. 121. Pieces of Pi (watch the entire video here)
  122. 122. Pieces of Pi   C E 1 of circumference 4 90 of circumference 360 CE   90 (watch the entire video here)
  123. 123. Pieces of Pi  119  C E 119 of circumference 360 CE  (watch the entire video here)
  124. 124. Pieces of Pi  x  C E   of circumference 360 2 360 CE x x r    arc length 2 360 x r (watch the entire video here)
  125. 125. Pieces of Pi O  C E  2 ofarea circof sect le's area 3 r 60 o 360 O x x C r E    ? (watch the entire video here)
  126. 126. Pieces of Pi  x  C E  2 ofarea circof sect le's area 3 r 60 o 360 O x x C r E    O  2 sector area 360 x r 360 x (watch the entire video here)
  127. 127. Pieces of Pi O 160 6 (watch the entire video here)
  128. 128. Pieces of Pi O 160  2 area 360 x r     2 6area 360 4 36 9 16 160       6 (watch the entire video here)
  129. 129. Pieces of Pi  x  C E  2 360 x CE r  x  C EO  2 area 360 x r (watch the entire video here)
  130. 130. Practice Question 20 A) 3 25 B) 3 25 C) 2 40 D) 3 50 E) 3      C B O Note: Figure not drawn to scale O is the center of the circle with radius 30. If x–w=20, what is the length of arc CDE ?  A E D w x y
  131. 131. 20 A) 3 25 B) 3 25 C) 2 40 D) 3 50 E) 3      C B O Note: Figure not drawn to scale O is the center of the circle with radius 30. If x–w=20, what is the length of arc CDE ?  A E D x  arc length 2 360 y r y 30 20x w  180x w  2 160 80 w w   80 80      arc length 2 360 2 60 9 8 4 3 0 0 0 3      Practice Question (watch the entire video here)
  132. 132. Circle Properties (watch the entire video here)
  133. 133.  Circle Properties A B x “x is an inscribed angle holding/containing chord AB”  “x is an inscribed angle holding/containing arc AB” (watch the entire video here)
  134. 134. Circle Properties  A B x x Inscribed angles holding the same chord/arc are equal x (watch the entire video here)
  135. 135. Circle Properties  A B x   C D x Inscribed angles holding chords/arcs of equal length are equal (watch the entire video here)
  136. 136. Circle Properties  An inscribed angle holding the diameter is a right angle (watch the entire video here)
  137. 137. Circle Properties    A B x O “Angle AOB is a central angle holding chord AB” 2x A central angle is twice as large as an inscribed angle holding the same chord/arc (watch the entire video here)
  138. 138. Circle Properties   The line from the center to the point of tangency is perpendicular to the tangent line “line l is tangent to the circle” (watch the entire video here)
  139. 139. Circle Properties * * * *  x 2x  (watch the entire video here)
  140. 140. Practice Question Note: Figure not drawn to scale  C x 20 D O B A A) 40 B) 50 C) 60 D) 70 E) 80 If is the center and , thenO AB CD x  E
  141. 141. Practice Question Note: Figure not drawn to scale C x D O B A A) 40 B) 50 C) 60 D) 70 E) 80 90 If is the center and , thenO AB CD x   A 10 20 90 90 80 80 E (watch the entire video here)
  142. 142. Volume & Surface Area (watch the entire video here)
  143. 143. Volume & Surface Area 1 ft 1 ft 1 ft3 1 ft  2 ft 3 ft 5 ft Volume length width height   3 Volume 2 3 5 30 ft     Volume (watch the entire video here)
  144. 144. Volume & Surface Area  r  height h 2 Volume r h  3 2 Volume r h 10   2 3Vo 1lume 90 0    Volume (watch the entire video here)
  145. 145. Volume & Surface Area Surface Area face • 6 faces • 12 edges • 8 vertices (watch the entire video here)
  146. 146. Volume & Surface Area Surface Area • 6 faces • 12 edges • 8 vertices edge edge edge edge (watch the entire video here)
  147. 147. Volume & Surface Area Surface Area • 6 faces • 12 edges • 8 vertices     vertex vertex vertex vertex vertex (watch the entire video here)
  148. 148. Volume & Surface Area Surface Area 8 cm 4 cm 5 cm 2 surface area 40 40 32 32 20 20 184 cm        (watch the entire video here)
  149. 149. Volume & Surface Area Surface Area 2 area r 2 area r h 2 r  area 2 2 r h rh        2 2 2 total area 2 2 2 2 r r rh r rh r r h               r h (watch the entire video here)
  150. 150. Volume & Surface Area length volume length width height   width height  r 2 volume r h   2 2 2 surface area 2 2 2 2 r r rh r rh r r h              surface area sum of areas of all 6 sides h (watch the entire video here)
  151. 151. Units of Measurement (watch the entire video here)
  152. 152. Units of Measurement • Metric: kilometers, kilograms, liters, etc. • English: miles, pounds, gallons, etc. What is the perimeter of this triangle? 12 13 (watch the entire video here)
  153. 153. Units of Measurement • If conversion is required, relationship will be given - e.g., (1 kilometer = 1000 meters) - e.g., (1 mile = 5280 feet) • Note: Relationships not given for units of time - e.g., (1 hour = 60 minutes) Conversions - e.g., (1 day = 24 hours) (watch the entire video here)
  154. 154. Geometry Data Sufficiency Questions (watch the entire video here)
  155. 155. Geometry Data Sufficiency Questions A B C x • Do not estimate lengths and angles (watch the entire video here)
  156. 156. Geometry Data Sufficiency Questions 1) 30x  2) AD DC What is the length of AD? B C D (watch the entire video here)
  157. 157. Geometry Data Sufficiency Questions 1) 30x  2) AD DC What is the length of AD? A B C D x • To find one length requires at least one other length (watch the entire video here)
  158. 158. Geometry Data Sufficiency Questions 1) 30x  2) AD DC What is the length of AD? INSUFFICIENT A B C D INSUFFICIENT 1&2) 30 &x AD DC  30 INSUFFICIENT E (watch the entire video here)
  159. 159. Geometry Data Sufficiency Questions 1) 10AC  2) 30x  If , what is the length of ?AE EC AB A B E x C D (watch the entire video here)
  160. 160. Geometry Data Sufficiency Questions 1) 10AC  2) 30x  If , what is the length of ?AE EC AB A B E x C D • Sketch figure and add information (watch the entire video here)
  161. 161. Geometry Data Sufficiency Questions 1) 10AC  2) 30x  If , what is the length of ?AE EC AB A B E x • Sketch figure and add information C D x 10 (watch the entire video here)
  162. 162. Geometry Data Sufficiency Questions 1) 10AC  2) 30x  If , what is the length of ?AE EC AB A B E x • Sketch figure and add information C D x 10 • Mentally grab and move points and lines (watch the entire video here)
  163. 163. Geometry Data Sufficiency Questions 1) 10AC  2) 30x  If , what is the length of ?AE EC AB A B E x • Sketch figure and add information C D • Mentally grab and move points and lines 10 x (watch the entire video here)
  164. 164. Geometry Data Sufficiency Questions 1) 10AC  2) 30x  If , what is the length of ?AE EC AB A B E x • Sketch figure and add information C D • Mentally grab and move points and lines 10 x (watch the entire video here)
  165. 165. Geometry Data Sufficiency Questions 1) 10AC  2) 30x  If , what is the length of ?AE EC AB A B E • Sketch figure and add information C D • Mentally grab and move points and lines INSUFFICIENT INSUFFICIENT 30 30 • To find one length requires at least one other length (watch the entire video here)
  166. 166. Geometry Data Sufficiency Questions 1) 10AC  2) 30x  If , what is the length of ?AE EC AB A B E C D 10 30 30 INSUFFICIENT INSUFFICIENT 1 & 2) 10 and 30AC x SUFFICIENT C (watch the entire video here)
  167. 167. Geometry Data Sufficiency Questions • Do not estimate lengths and angles • To find one length, requires at least one other length • Sketch diagram and add information • Mentally grab and move points and lines (watch the entire video here)
  168. 168. Geometry Strategies – Part II (watch the entire video here)
  169. 169. • Redraw figures • Add all given information • Add any information that can be deduced • Add/extend lines • Assign variables and use algebra • Problem solving questions drawn to scale: • Circle: • Break areas/volumes into manageable pieces • Two or more triangles and length required • Right triangle: - use Pythagorean Theorem to relate sides - watch for Pythagorean Triples and special triangles - beware of circle properties (inscribed/central angles, tangent lines) - look for isosceles triangles - estimate to confirm calculations and guide guesses - look for similar triangles Geometry Strategies – Part II (watch the entire video here)
  170. 170. Practice Question 1 2 Are lines l1 and l2 parallel? 2) b d a b c d e 1) 180e b  A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient D) EACH statement ALONE is sufficient E) Statements (1) and (2) TOGETHER are NOT sufficient
  171. 171. Practice Question 2) b d 1) 180e b          1 2 a b c d e 180   1 2      SUFFICIENT SUFFICIENT D  Are lines l1 and l2 parallel? (watch the entire video here)
  172. 172. Practice Question Note: Figure not drawn to scaleA) 1 4 B) 3 3 C) 2 5 D) 3 5 E) 2 60 1x  4 3x  What is the value of x ?
  173. 173. Practice Question Note: Figure not drawn to scaleA) 1 4 B) 3 3 C) 2 5 D) 3 5 E) 2 30 60 1 2 3 60 1x  4 3x  What is the value of x ? 30 With similar triangles, the ratio of any pair of corresponding sides is the same    1 4 3 1 2 2 1 1 4 3 2 2 4 3 2 2 3 5 2 5 2 x x x x x x x x x              (watch the entire video here)
  174. 174. Practice Question Note: Figure not drawn to scale If is tangent to the circle with center , thenAC O DBC   D O B CA  40 A) 50° B) 55° C) 60° D) 65° E) 70°
  175. 175. Practice Question Note: Figure not drawn to scale If is tangent to the circle with center , thenAC O DBC   D O B CA  40 A) 50° B) 55° C) 60° D) 65° E) 70° 50 130 25 25 65 (watch the entire video here)
  176. 176. Practice Question B A C D 2) AC CD 1) 5BC  If 12, does 90 ?AC ACB   A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient D) EACH statement ALONE is sufficient E) Statements (1) and (2) TOGETHER are NOT sufficient
  177. 177. Practice Question 2) AC CD 1) 5BC INSUFFICIENT B If 12, does 90 ?AC ACB   A C D 12 INSUFFICIENT 1 & 2) 12 5 INSUFFICIENT E (watch the entire video here)
  178. 178. Practice Question What is the area of triangle ?ABC 60 5 12 Note: Figure not drawn to scale A) 15 B) 15 3 5 119 C) 2 D) 32.5 E) 36 A BC
  179. 179. Practice Question What is the area of triangle ?ABC 60 Note: Figure not drawn to scale A) 15 B) 15 3 5 119 C) 2 D) 32.5 E) 36 enlargement factor: 6 12  3 6 36h   6 3 5 A BC base height area 2   5 6 3 area 2 30 3 2 15 3     (watch the entire video here)
  180. 180. Practice Question If is a parallelogram, then what is its perimeter?ABCD Note: Figure not drawn to scale A B CD 3 3x y  4 2 2y x  6x y  2 6 13x y  A) 22 B) 24 C) 26 D) 28 E) 30
  181. 181. Practice Question If is a parallelogram, then what is its perimeter?ABCD Note: Figure not drawn to scale A) 22 B) 24 C) 26 D) 28 E) 30             perimeter 3 3 2 6 13 4 2 2 1 6 4 4 18 4 18 4 18 22 x y x y y x x y x x y y                      A B CD 6 2 6 13 5 7 x y x y x y         6x y  2 6 13x y  3 3 4 2 2 5 1 5 5 x y y x x x y y          4 2 2y x  3 3x y  (watch the entire video here)
  182. 182. Practice Question What is the value of ?x Note: Figure not drawn to scale  155 3x  6 30x   4 70x  A) 5 B) 7 C) 15 D) 21 E) 25
  183. 183. Practice Question What is the value of ?x Note: Figure not drawn to scale  155 3x  6 30x   4 70x  A) 5 B) 7 C) 15 D) 21 E) 25  180 4 70x     180 155 3x     6 30x          180 4 70 180 155 3 6 30 180 110 4 25 3 6 30 180 105 5 180 5 75 15 x x x x x x x x x                               (watch the entire video here)
  184. 184. Practice Question K is the surface area of cylinder A. If the radius of cylinder B is twice the radius of cylinder A, and the height of cylinder B is twice that of cylinder A, what is the surface area of cylinder B? A) 2K B) 3K C) 4K D) 6K E) 8K
  185. 185. Practice Question K is the surface area of cylinder A. If the radius of cylinder B is twice the radius of cylinder A, and the height of cylinder B is twice that of cylinder A, what is the surface area of cylinder B? A) 2K B) 3K C) 4K D) 6K E) 8K 2 surface area 2 2r rh   1 1      2 2 2 2 2 1 1 1 4            2 2 2 surface area 2 2r rh        2 2 2 8 8 1 2 2 2 6           A B K 4K (watch the entire video here)
  186. 186. Practice Question Note: Figure not drawn to scale 2) AC AB 1) 8CB  C B A x If the circle has radius 4, is 80?x 
  187. 187. Practice Question Note: Figure not drawn to scale 2) AC AB 1) 8CB  C B A x If the circle has radius 4, is 80?x  SUFFICIENT INSUFFICIENT A (watch the entire video here)
  188. 188. Practice Question 2) BE EA 1) 30BCE  If ABCD is a rectangle, is the area of ∆EBC greater than the area of ∆AEC ? C B AD E
  189. 189. Practice Question 2) BE EA 1) 30BCE  C B AD E If ABCD is a rectangle, is the area of ∆EBC greater than the area of ∆AEC ? B E A DC harea 2 bh  Which triangle has the longest base?INSUFFICIENT SUFFICIENT B (watch the entire video here)
  190. 190. Practice Question Note: Figure not drawn to scale 2 1) 14 48 0y y   A C B 55 y hat is the area of ?W ABC 2 2) 16 60 0y y   A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient D) EACH statement ALONE is sufficient E) Statements (1) and (2) TOGETHER are NOT sufficient
  191. 191. Practice Question Note: Figure not drawn to scale 2 1) 14 48 0y y   A C B 55 y hat is the area of ?W ABC 2 2) 16 60 0y y      2 1) 14 48 0 6 8 0 6, 8 y y y y y        area 12 area 12 SUFFICIENT h    2 2) 16 60 0 6 10 0 6, 10 y y y y y        area 12 The sum of the lengths of any two sides of a triangle must be greater than the third side. 5 5 10   SUFFICIENT D (watch the entire video here)
  192. 192. Practice Question Note: Figure not drawn to scale A B D C E F If bisects , and bisects , thenBD CBE DE BEF w   w 50 A) 25 B) 35 C) 50 D) 55 E) 65
  193. 193. Practice Question Note: Figure not drawn to scale A B D C E F If bisects , and bisects , thenBD CBE DE BEF w   w 50 x x y y 180 2y  180 2x       50 180 2 180 2 180 410 2 2 180 230 2 2 23 5 0 11 2 x y x y x y x x y y              A) 25 B) 35 C) 50 D) 55 E) 65   180 180 180 x w x y w x y y w          11180 65 5   (watch the entire video here)
  194. 194. Practice Question Note: Figure not drawn to scale If is a rectangle, then what is the length of ?ABCD EC A) 7.8 B) 8 C) 8.4 D) 9 E) 9.6 A B D C E 12 16
  195. 195. Practice Question Note: Figure not drawn to scale If is a rectangle, then what is the length of ?ABCD EC A) 7.8 B) 8 C) 8.4 D) 9 E) 9.6 A B D C E B C DE D C B E 12 16 16 16 121216 20 area 2 bh    12 16 area 9 2 6   h area 2 bh  20 2 96 10 .6 9 9 6 h h h    EC (watch the entire video here)
  196. 196. Practice Question If the both circles have radius 6, and O and P are their centers, what is the area of the shaded region? A) 24 18 3 B) 24 12 3 C) 18 D) 36 24 3 E) 18 12 3           PO 
  197. 197. Practice Question If the both circles have radius 6, and O and P are their centers, what is the area of the shaded region? A) 24 18 3 B) 24 12 3 C) 18 D) 36 24 3 E) 18 12 3          ca b  2 6 60 6 360 6 6 6 a b d e b c d f                 2 sector area 360 x r O  e P f d               24 24 24 24 24 24 1 9 3 3 8 9 3 b d b d b d a b d e b c d f a b c d e f a b c d e f a b c d e f a b c d e f                                                 b 66 6   2 3 side area 4   2 3 b 3 6 9 4    b d a b c d e f     (watch the entire video here)
  198. 198. GMAT Geometry - Everything you need to know For additional practice questions, see the bottom of our Geometry module www.GMATPrepNow.com
  199. 199. GMAT Geometry - Everything you need to know If you enjoyed this unique learning format, let us know, and we’ll add similar resources to our SlideShare page

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