GMAT (Fach) / Quantitative - Geometry (Lektion)

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• quadrilaterals: an overview parallelogram: opposite side + angels equal trapezoid: one pair of opposite sides is parallel rectangle: opposing sites are equal, all angles are 90 degrees rhombus: all sides equal, opposite angles equal square: all angles are 90 degrees, all sides equal
• Sum of Interior Angles of a Polygon (n-2) * 180 = sum of interior angles of a polygon
• Area of RECTANGLE & PARALLELOGRAM Area of a triangle: Base*Height / 2 Area of a RECTANGLE: Length * Width Area of a  TRAPEZOID = ((Base 1 + Base 2) * Height ) / 2 Area of a PARALLELOGRAM = Base * Height Area of a RHOMBUS = (Diagonal 1 + Diagonal 2) / 2
• Triangel, Trapezoid, Rhombus TRIANGLE:          Base* h / 2 TRAPEZOID:     ((Base 1 + Base 2) * h) / 2 RHOMBUS:        (Diagonal 1 + Diagonal 2) / 2
• Geometry on a star 1. Detrmine sum of interior angles: 180 * (5-2) = 540 2. Find the n traingles within the star and set up equations a + b + c = 180  3. Sum the 5 equations together and solve
• A right circular cone radius = height = 1:1, Isosceles traingle
• THEORY: Key Properties of triangles THE SUM OF ALL ANGLES = 180 ANGLES CORRESPOND TO THEIR OPPOSITE SIDE  largest angle -> largest opposite side smallest angle -> smallest opposite side THE SUM OF ANY TWO SIDES MUST BE GRESTER THAN THE 3RD SIDE 1<X<7
• Equal Triangles if the two sides are in the ratio A:B then Areas will be in ratio A2:B2
• Shortcut for Traingles 3 -  4 -  5      +multiples 5 - 12 - 13    +multiples 8 - 15 - 17    +multiples
• Isosceles triangle 45      :         45         :          90 x        :           x         :          x√2 -> diagonal of square / cube  x√2 / x√3
• THE 30:60:90 Triangle 30      :          60        :          90 x        :          x√3         :        2x
• THEORY: CIRCLE CIRCUMFERENCE C = π * d
• THEORY: Area of a circle A = π * r2
• Volume Cylinder V = π r2 h
• THEORY: CYLINDERS Surface Area S = 2 * (π * r2) + 2 (π*r) * h S = 2 (Area of circle) + 2 (perimeter of circle)* height
• THEORY: Inscribed Triangles If one side of a traingle is the diameter of a circle, then the triangle MUST be a right triangle. Any RIGHT triangle in a circle must have d as one of its sides.
• THEORY: Inscribed Angles The Inscribed angle is equal to half of the arc it intercepts 
• THEORY: Calculate the arc length Calculate the Circumference : π*d Calculate the fraction: x°/360° *π*d
• THEORY: The perimeter = Umfang Calculate the sum of n sides
• Triangle ABC is equilateral and point P is equidistant from verticles A,B and C. If triangle ABC is rotated clockwise about point P, what is the minimum number of degrees the triangle must be rotated so that B will be in the position where A is now? Since ABC is equilateral, the measure of ACB is 60.  The measure of BCD is 180 - 60 = 120 degrees. Rotating the figure clockwise about point P through an angle of 120 degrees will produce the figure admired.
• RATIO: Side length of an equilateral triangle 1:√3:2 30:60:90
• The interior of a rectangular carton hat a volume of x cubs feet at a ratio 3:2:2. Which of the following equals the height of the carton? Let c represent the constant of proportionality.  The volume of the carton is then 3c*2c*2c = 12c3  12c3 = x Solve for c!
• Circle calculations: THe circle with enter C shown above is tangent to 2 axes. If the hypothenuse is the line OC, what is the radius of the circle in terms of k? In a circle, all distances from the circle to the center are the same, hence  a2 + b2 = c2  r2 + r2 = k2 2 r2=k2
• Three solid cheese balls with diameters of 2,4 and 6 inches were combined to form a single cheeseball. What was the approx. diameter of the new cheeseball if its volume is 4/3 pi r^3? Using V = 4/3 pi*r3 = 4/3 pi * (13+23+33) = 4/3 pi * 36 R is the radius of the new cheeseball, thus:  V = 4/3 pi*36 = R3 = 36, R is 3√36 and the diameter is 23√36
• A parallelogram shown has four sides of equal length. What is the ratio of the length of the shorter diagonal to the length of the longer diagonal? Parallelogram = two equilateral triangles =  four 30:60:90 triangle.  30 :      60      :     90  x    :     x√3     :     2x SHORTER = 2*x  LONGER = 2* x√3 Ratio: 1/ x√3
• The hypotenuse of a triangle is 10 cm. What is the perimeter of the triangle? 1) The Area is 25 cm 2) The 2 legs of the circle are of equal length EACH statement ALONE is sufficient. Look for: a+b given that the hypotenuse is 10 cm, a2+b2= 100 1) a*b/2 = 25 ⇒ a*b = 50  a2+b2= 100  (a+b)2 = a2+b2 + 2ab = 100+ 2*50  (a+b)2 = 200  a+b = √200 SUFFICIENT 2) 2a2 = 100 a2 = 50 a = √50 perimeter = 2a= 2√50
• D, E and F are points on the sides of triangle ABC above, such that quadrilateral AEFD is a rectangle. If DC=1, and CF=4, what is the value of AD? (1) 3EB = AB (2) FB = 2 Note that the figure actually contains three similar right triangles: ∆DCF and ∆EFB are both right triangles, and each shares an additional angle with large triangle ACB (∠C in DCF and ∠B in EFB). Thus, the third angle in each of the triangles must also equal the respective third angle in ∆ABC, and the triangles are similar. (1) 3EB = AB (2) FB = 2 (1): form the proportion between ∆ABC and ∆EBF as follows: {Long leg ratio} = {short leg ratio} AC/EF = AB/EB From the question stem, AC = x + 1.  AB = 3EB (x + 1)/x = 3EB/EB = 3. -->    x + 1 = 3x SUFFICIENT (2): form the proportion between ∆ABC and ∆EBF as follows: {hypotenuse ratio}  = {long leg ratio} AC/EF = BC/BF From the question stem, AC = x + 1. From stat. (2), BC = CF + BF = 4 + 2 = 6. Plug these values into the proportion: (x + 1)/x = 6/2 =  3. -->    x + 1 = 3x From here it is possible to find a single value for x. Therefore, SUFFICIENT D
• DS: The surface area of cylinder A is how many times the surface area of cylinder B? (1) The diameter of A is twice the diameter of B. (2) The height of A is equal to the height of B. (1) The diameter of A is twice the diameter of B. (2) The height of A is equal to the height of B. Stat. (1): You only know the relationship between the radii of the two cylinders. You do not know anything about the heights of the two cylinders. Stat. (1)->IS->BCE.       Stat. (2): You only know that the heights of the two cylinders are equal. You do not know anything about the radii of the two cylinders. Stat. (2)->IS->CE. Stat. (1) + (2) Combined, you get the relationship between the radii as well as heights of the two cylinders. Let 2r be the radius of cylinder A and r be the radius of cylinder B as per Stat. (1) and let h be the height of both cylinders as the height of cylinder A is equal to that of cylinder B. By plugging in these in the formula of the surface area of a cylinder, you find that the surface area of cylinder A = 8 π r² + 4 π r h and that of cylinder B = 2 π r² + 2π r h. It is not possible to determine the ratio between the surface areas of cylinder A and B without knowing the specific values of r and h. Stat.(1) + (2)->IS->E.    
• Are of equilateral traingle s2√3 / 4
• Diagonal of a Square d = s√2
• Diagonal of a rectangle d= s√3
• Volume Circle / Ball V = 4/3 ∏*r3

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