Math-Book-GMAT-Club

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• The midsegment is always parallel to the third side of the triangle.• The midsegment is always half the length of the third side.• A triangle has three possible midsegments, depending on which pair of sides is initially joined.Relationship of sides to interior angles in a triangle• The shortest side is always opposite the smallest interior angle• The longest side is always opposite the largest interior angleAngle bisector An angle bisector divides the angle into two angles with equal measures.• An angle only has one bisector.• Each point of an angle bisector is equidistant from the sides of the angle.• The angle bisector theorem states that the ratio of the length of the line segment BD to the length ofsegment DC is equal to the ratio of the length of side AB to the length of side AC:• The incenter is the point where the angle bisectors intersect. The incenter is also the center of thetriangle's incircle ‐ the largest circle that will fit inside the triangle.Similar Triangles Triangles in which the three angles are identical.• It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles aresimilar; the third set will be identical because all of the angles of a triangle always sum to 180º.• In similar triangles, the sides of the triangles are in some proportion to one another. For example, a triangle withlengths 3, 4, and 5 has the same angle measures as a triangle with lengths 6, 8, and 10. The two triangles aresimilar, and all of the sides of the larger triangle are twice the size of the corresponding legs on the smallertriangle.• If two similar triangles have sides in the ratio, then their areas are in the ratio‐ 59 ‐GMAT Club Math Bookpart of GMAT ToolKit iPhone App
Congruence of triangles Two triangles are congruent if their corresponding sides are equal in lengthand their corresponding angles are equal in size.1. SAS (Side‐Angle‐Side): If two pairs of sides of two triangles are equal in length, and the included angles areequal in measurement, then the triangles are congruent.2. SSS (Side‐Side‐Side): If three pairs of sides of two triangles are equal in length, then the triangles arecongruent.3. ASA (Angle‐Side‐Angle): If two pairs of angles of two triangles are equal in measurement, and the includedsides are equal in length, then the triangles are congruent.So, knowing SAS or ASA is sufficient to determine unknown angles or sides.NOTE IMPORTANT EXCEPTION:The SSA condition (Side‐Side‐Angle) which specifies two sides and a non‐included angle (also known as ASS, orAngle‐Side‐Side) does not always prove congruence, even when the equal angles are opposite equal sides.Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter than theknown adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. Inall other cases with corresponding equalities, SSA proves congruence.The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also knownas the HL (Hypotenuse‐Leg) condition or the RHS (Right‐angle‐Hypotenuse‐Side) condition), we can calculate thethird side and fall back on SSS.To establish congruence, it is also necessary to check that the equal angles are opposite equal sides.So, knowing two sides and non‐included angle is NOT sufficient to calculate unknown side and angles.Angle‐Angle‐AngleAAA (Angle‐Angle‐Angle) says nothing about the size of the two triangles and hence proves only similarity and notcongruence.So, knowing three angles is NOT sufficient to determine lengths of the sides.Scalene triangle all sides and angles are different from one another• All properties mentioned above can be applied to the scalene triangle, if not mentioned the special cases(equilateral, etc.)‐ 60 ‐GMAT Club Math Bookpart of GMAT ToolKit iPhone App
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For the latest version of the GMAT
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Number TheoryDefinitionNumber Theor
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• 1 (and ‐1) are divisors of ev
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other digits: 21‐(9+8+6+9)=‐11,
Page 10 and 11: Sum of n first positive integers:Su
Page 12 and 13: Converting Decimals to Fractions•
Page 14 and 15: Operations involving the same bases
Page 16 and 17: PERCENTDefinitionA percentage is a
Page 18 and 19: Absolute ValueDefinitionThe absolut
Page 20 and 21: satisfied (‐15 is not within (‐
Page 22 and 23: FactorialsDefinitionThe factorial o
Page 24 and 25: AlgebraScopeManipulation of various
Page 26 and 27: There is a common misconception tha
Page 28 and 29: RemaindersDefinitionIf and are posi
Page 30 and 31: A. 2B. 4C. 8D. 20E. 45divided by yi
Page 32 and 33: Example #10 (hard)When positive int
Page 34 and 35: Word Problems OverviewThe Following
Page 36 and 37: Key words: less than (implies subtr
Page 38 and 39: Distance/Speed/Time Word ProblemsWh
Page 40 and 41: In the next section we will learn h
Page 42 and 43: Time(3) = Time(1) + Time(2) ‐‐
Page 44 and 45: 3*x = 5*(x ‐ 20) ‐‐> x = 50 m
Page 46 and 47: Work Word ProblemsWhat is a ‘Work
Page 48 and 49: Example 3:Working together, printer
Page 50 and 51: Advanced Overlapping Sets ProblemsS
Page 52 and 53: the writing club. If 6 students sig
Page 54 and 55: two of the courses and 1 employee h
Page 56 and 57: Consider this problem to be an over
Page 58 and 59: Median The median of a triangle is
Page 62 and 63: Equilateral triangle all sides have
Page 64 and 65: • A right triangle where the angl
Page 66 and 67: Then the triangles ABC, CHB and CHA
Page 68 and 69: Quadrilateral A polygon with four '
Page 70 and 71: Squares A 4‐sided regular polygon
Page 72 and 73: • Area ‐ The usual way to calcu
Page 74 and 75: • A circle is the shape with the
Page 76 and 77: ChordA line that links two points o
Page 78 and 79: angle.• An inscribed angle is exa
Page 80 and 81: chords.‐ This becomes the theorem
Page 82 and 83: Coordinate GeometryDefinitionCoordi
Page 84 and 85: As you can see, the distance formul
Page 86 and 87: Lines in Coordinate GeometryIn Eucl
Page 88 and 89: The equation of a straight line tha
Page 90 and 91: Positive slopeHere, y increases as
Page 92 and 93: A horizontal line has a slope of ze
Page 94 and 95: Solution: We first find the slope o
Page 96 and 97: To answer, we must find the slope o
Page 98 and 99: Example #2Q: Find the point of inte
Page 100 and 101: ParabolaA parabola is the graph ass
Page 102 and 103: Standard DeviationDefinitionStandar
Page 104 and 105: Standard deviation of is . Decrease
Page 106 and 107: ProbabilityDefinitionA number expre
Page 108 and 109: Therefore, the probability of getti
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2) Reversal combinatorial approach:
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Combinations & PermutationsDefiniti
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1. The total number of arrangements
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is the first termDefining Propertie
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Solution(1) a1=8, does not tell us
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CuboidA cube is the 3‐D generaliz
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ConeA cone is a 3‐D object obtain
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Volume =Surface Area w/o base =Surf
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(2) : Sum of sides = . Not sufficie
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Math-Book-GMAT-Club

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