Geometry¶

• Line
• Bisector: divide into 2
• Perpendicular bisector
• Transversal: passes through 2 || lines
  • All angles formed are equal due to VOA and AIA
• Angle
• GRE only focuses on degrees
  • No need to learn radians
• Vertically-opposite angles are equal
• Alternate interior angles are equal

Triangles¶

• Vertices
• Angles add upto 180
• Edges
• Angles-Edges
  • Shortest edger oppo to smallest angle
  • Longest edge oppo to largest angle
• \(\vert a-b \vert < c < (a+b)\)

Area = \(\dfrac{1}{2} bh\)

Isosceles¶

• 2 equal angles
• 2 equal edges

Equlaterial¶

• 3 equal angles: 60,60,60
• 3 equal edges

Area = \(\dfrac{\sqrt{3}}{4} a^2\)

Altitudes of isosceles and equilateral triangles always bisect the base

Right-Triangle¶

• Pythagorean theorem: \(a^2+b^2 = c^2\)

• Pythagorean triplet: Set of 3 integers that can be the sides of a right triangle

  • Common
  • 3-4-5
  • 5-12-13
  • 8-15-17
  • 7-24-25
  • Multiple of triples is also a triple
  • 2 corresponding sides are required for a triple

• 45-45-90: \(x, x, x \sqrt{2}\)

• Hiding in squares
• 30-60-90: \(x, x\sqrt{3}, 2x\)
• Hiding in equilateral triangle

Similar Triangles¶

All three angles are equal

Ratio of any pair of corresponding sides is the same

Quadrilaterals¶

4 edges

Angles add upto 360 deg

• Parallelogram
• Opposite sides are parallel
• Opposite sides are equal
• Opposite angles are equal
• Rhombus
• Parallelogram with equal edges
• Diagonals are perpendicular bisectors
• \(\dfrac{1}{2} d_1 d_2\)
• Rectangle
• Parallelogram with all angles 90
• Diagonals are equal length
• Square
• Rectangle with equal edges
• Diagonals are perpendicular bisectors
• Diagonals are equal length
• Trapezoid/Trapezium
• 1 pair of parallel edges
• \(\dfrac{1}{2}h(a+b)\)

Polygons¶

Sum of angles of polygon with \(n\) edges = \(180(n-2)\)

Convex polygon: all interior angles < 180

• Pentagon: angles add up to 540
• Hexagon: angles add up to 720

Regular polygon: equal sides and equal angles

Circles¶

• center
• radius
• Chord: line segment passing connecting 2 points of circumference
• diameter: chord passing through center
• Arc
• Minor arc
• Major arc
• Sector
• Circumference: \(2 \pi r\)
• Area: \(\pi r^2\)

Properties

• Inscribed angles on the same side of a chord/arc held are equal
• Inscribed angles holding chord/arc of equal lengths are equal
• Inscribed angles holding diameter is 90deg
• Central angle is 2 x inscribed angle holding the same chord/arc
• Radius \(\perp\) tangent at the point of intersection

Volume¶

• Cube: \(a^3\)
• Cuboid: \(lbh\)
• Cylinder: \(\pi r^2 h\)

Surface Area¶

• Cube: \(6 a^2\)
• Cuboid: \(2(lb + bh + lh)\)
• Cylinder: \(2 \pi r^2 + 2 \pi r h\)

Units of Measurement¶

• Metric: km, kg, L
• English/Imperial: miles, pounds, gallons

All conversions will be given in GRE

Only units of time are expected to be known

Strategies¶

• Redraw figures
• Add all given information
• Add all information that can be deduced
• Add/extend lines
• Assign vars and use algebra
• 2/more triangles & lengths required
• Look for similar triangles
• Right triangle
• Look out for triples
• Use pythagorean theorem
• Circles
• Look out for circle properties
• Look out for isosceles triangles

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