gmat geometry

  1. area of a triangle
    (base x height)/2
  2. area of a trapezoid
    Image Upload 1
  3. area of any parallelogram 
    base x height
  4. area of a rhombus
    Image Upload 2
  5. surface area
    • the sum of all the faces
    • to determine the SA of a rectangular solid, you must find the area of each face
    • to determine the SA of a cube, you only need the length of one side
  6. volume
    length x width x height
  7. parallelogram
    opposite sides and opposite angles are equal
  8. trapezoid
    one pair of opposite sides is parallel.  
  9. rhombus
    all sides are equal.  opposite angles are equal.
  10. sum of interior angles of a polygon
    (n-2) x 180 where n = the number of sides
  11. isosceles triangle
    • if two sides are equal, their opposite angles are also equal
    • 45-45-90
    • leg:leg:hypotenuse 
    • 1:1:Image Upload 3
    • x:x:xImage Upload 4
  12. triangle inequality law
    • the sum of any two sides of a triangle must be great than the third side
    • the difference of any two sides must be less than  the third side
  13. pythagorean theorem
    a2 + b2 = c2
  14. common right triangles
    • 3-4-5; 32 + 42 = 52 (9 + 16 = 25); key multipliers: 6-8-10, 9-12-15, 12-16-20
    • 5-12-13; 52 + 122 = 132 (25 + 144 = 169); key multipliers: 10-24-26
    • 8-15-17; 82 + 152 = 172 (64 + 225 = 289)
  15. equilateral triangle
    • all three sides (and all three angles) are equal
    • each angle is 60 degrees
    • two 30-60-90 triangle form an equilateral triangle
  16. 30-60-90
    • short leg : long leg : hypotenuse
    • 1 : Image Upload 5 : 2
    • x : xImage Upload 6: 2x
  17. diagonal of a square
    • sImage Upload 7 where s is an side of the cube
    • also the face diagonal of a cube
  18. diagonal of a cube
    sImage Upload 8 where s is an edge of the cube
  19. diagonal of a rectangle 
    find either the length and the width or one dimension and the proportion of one to the other
  20. deluxe pythagorean theorem
    • used to find length of the main diagonal of a rectangular solid
    • d2 = x2 + y2 + z2, where x, y, and z are the sides of the rectangular solid and d is the main diagonal
    • no formula approach: find diagonal of the bottom face and use this as the base leg of another right triangle (see page 4.35)
  21. similar triangles
    • all corresponding angles are equal and their corresponding sides are proportion
    • if two similar triangles have corresponding side lengths in ratio a:b, then their ares will be in ratio a2:b2
  22. area of an equilateral triangle with a side length of S
    • Image Upload 9
  23. area of a right triangle
    (1/2) Hypotenuse x height from hypotenuse
  24. radius
    • any line segment that connects the center point to a point on the circle
    • half the distance across a circle
  25. chord
    any line segment that connects two points on a circle
  26. diameter
    • any chord that passes through the center of the circle
    • 2 x r
  27. circumference
    • Image Upload 10d
    • 2Image Upload 11r
    • revolution (i.e. a full turn of a spinning wheel)
  28. area of a circle
    Image Upload 12r
  29. arc length
    • corresponds to the crust
    • first find the circumference of the circle
    • then, use the central angle to determine what fraction the arc is of the entire circle
    • multiple fraction and C
  30. perimeter of a sector
    • corresponds to a slice of pizza
    • formed by the arc and two radii (sum arc length and two radii)
  31. area of a sector of a circle
    • first, find the area of the entire circle
    • then, use the central angle to determine what fraction of the entire circle is represented by the sector
    • multiply fraction and area
  32. central angle
    an angle whose vertex lies at the center point of a circle
  33. inscribed angle
    • has its vertex on the circle itself
    • equal to half of the arc it intercepts
  34. inscribed triangles
    • a triangle is said to be inscribed in a circle if all of the vertices of the triangle are points on the circle
    • if one of the sides of an inscribed triangle is a diameter of the circle, then the triangle must be a right triangle
  35. surface area of a cylinder
    2Image Upload 13r2 + 2Image Upload 14rh
  36. volume of a cylinder
    Image Upload 15r2h
  37. three properties of intersecting lines
    • interior angles form a circle; sum of these angles is 360 degrees
    • interior angles that combine to form a line sum to 180 degrees (supplementary angles)
    • angles found opposite each other where these two lines intersect are equal (vertical angles)
  38. exterior angles of a triangle
    equal to the sum of the two non-adjacent (opposite) interior angles of the triangle
  39. parallel lines cut by a transversal
    • all acute angles are equal
    • all obtuse angles are equal
    • any acute angle is supplementary to any obtuse angle
  40. slope of a line
    rise over run
  41. 4 types of slopes
    • positive
    • negative
    • zero (----)
    • undefined slop (l)
  42. intercept (x- and y-intercept)
    • a point where a line intersects a coordinate axis
    • x-intercept is the point on the line at which y=0
    • y-intercept is the point on the line at which x=0
    • to find x-intercept, plug in 0 for y
    • to find y-intercept, plug in 0 for x
  43. slope-intercept equation
    y = mx + b
  44. horizontal lines
    y = some number
  45. vertical lines
    x = some number
  46. finding distance between 2 points
    use pythagorean theorem
  47. Data Sufficiency Answer Choices
    • (A) Statement 1 ALONE is sufficient, but statement 2 is NOT sufficient
    • (B) State 2 ALONE is sufficient, but statement 1 is NOT sufficient
    • (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
    • (D) EACH statement ALONE is sufficient
    • (E) Statement 1 and 2 TOGETHER are NOT sufficient
  48. The Data Sufficiency Process
    • Separate additional info from the actual question
    • determine whether the question is Value or Yes/No
    • Decide exactly what the question is asking
    • Use the Grid to Evaluate the statements
    • AD
    • BCE
  49. General Geometry Approach
    • Draw or redraw the figure
    • fill in the given information
    • Identify the wanted element
    • Infer from the givens
    • Find the wanted element
  50. Unspecified number - picking numbers
    (e.g. If the length of the side of a cube decreases by one-half, by what percentage will the volume of the cube decreases?)
    • Infer cube has a side of 2 unites ("smart" number bc it is divisible by 2 - the denominator of one-half)
    • volume = 2 x 2 x 2 = 8
    • if cube decreases by one-half, its new length is 2 - .5 (2) = 1 unit
    • its new volume = 1 x 1 x 1 = 1
    • Percent decreases = change/original
    • (8-1)/8 = 7/8 = 0.875 or 87.5%
  51. similar triangles (2)
    any time two triangles each have a right angle and also share an additional right angle, they will be similar
  52. maximum area of a quadrilateral
    • of all quadrilaterals with a given perimeter, the square has the largest area
    • of all quadrilaterals with a given area, the square has the minimum perimeter
  53. maximum area of a parallelogram or triangle
    • if you are given two sides of a triangle or parallelogram, you can maximize the area by placing those two sides perpendicular to each other
    • rule holds for rhombuses as well
  54. key questions about parabola:
    (1) how many times does the parabola touch the x-axis?
    (2) If the parabola does touch the x-axis, where does it touch?
    • In other words, the parabola touches the x-axis at those values of x that makes f(x)=0
    • You can solve for zero by factoring and solving the equation directly. 
    • You might plug in points and draw the parabola
    • Use quadratic formula to quickly tell how many solutions the equation has by looking at the discriminate (b^2 = 4ac)
    • Image Upload 16

    • (1) If b^2 - 4ac > 0, two roots --> parabola crosses the x-axis twice and has two x-intercepts
    • (2) If b^2 - 4ac = 0, produce one root, parabola touches the x-axis once and has just one x-intercept
    • (3) If b^2 - 4ac < 0, square root operation cannot be performed, produces no roots, parabola never touches the x-axis (it has no x-intercept)
  55. Perpendicular Bisectors (rare on GMAT)
    • forms a 90 degrees angle with the segment and divides the segment exactly in half 
    • the perpendicular bisector has the negative reciprocal slope of the line segment it bisects
    • (1) Find the slope of segment AB
    • (2) Find the slope of the perpendicular bisector of AB
    • (3) Find the midpoint of AB
    • midpoint between point A(x1, y1) and point B(x2, y2) is (Image Upload 17Image Upload 18)
    • (4) Put the information together.
  56. The Intersection of Two Lines 
    ex. at what point does the line represented by y-4x-10 intersect the line represented by 2x+3y=26
    • replace y in the second equation with 4x-10 and solve for x
    • solve for y
Card Set
gmat geometry
basic geometry flashcards