gmat geometry
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area of a triangle
(base x height)/2


area of any parallelogram
base x height


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

volume
length x width x height

parallelogram
opposite sides and opposite angles are equal

trapezoid
one pair of opposite sides is parallel.

rhombus
all sides are equal. opposite angles are equal.

sum of interior angles of a polygon
(n2) x 180 where n = the number of sides

isosceles triangle
 if two sides are equal, their opposite angles are also equal
 454590
 leg:leg:hypotenuse
 1:1:
 x:x:x

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

pythagorean theorem
a^{2} + b^{2} = c^{2}

common right triangles
 345; 3^{2} + 4^{2} = 5^{2} (9 + 16 = 25); key multipliers: 6810, 91215, 121620
 51213; 5^{2} + 12^{2} = 13^{2} (25 + 144 = 169); key multipliers: 102426
 81517; 8^{2} + 15^{2} = 17^{2} (64 + 225 = 289)

equilateral triangle
 all three sides (and all three angles) are equal
 each angle is 60 degrees
 two 306090 triangle form an equilateral triangle

306090
 short leg : long leg : hypotenuse
 1 : : 2
 x : x: 2x

diagonal of a square
 s where s is an side of the cube
 also the face diagonal of a cube

diagonal of a cube
s
where
s is an edge of the cube

diagonal of a rectangle
find either the length and the width or one dimension and the proportion of one to the other

deluxe pythagorean theorem
 used to find length of the main diagonal of a rectangular solid
 d^{2} = x^{2} + y^{2} + z^{2}, 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)

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 a^{2}:b^{2}

area of an equilateral triangle with a side length of S

area of a right triangle
(1/2) Hypotenuse x height from hypotenuse

radius
 any line segment that connects the center point to a point on the circle
 half the distance across a circle

chord
any line segment that connects two points on a circle

diameter
 any chord that passes through the center of the circle
 2 x r

circumference
 d
 2r
 revolution (i.e. a full turn of a spinning wheel)

area of a circle
r
^{2 }

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

perimeter of a sector
 corresponds to a slice of pizza
 formed by the arc and two radii (sum arc length and two radii)

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

central angle
an angle whose vertex lies at the center point of a circle

inscribed angle
 has its vertex on the circle itself
 equal to half of the arc it intercepts

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

surface area of a cylinder
2
r
^{2} + 2
rh

volume of a cylinder
r
^{2}h

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)

exterior angles of a triangle
equal to the sum of the two nonadjacent (opposite) interior angles of the triangle

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

slope of a line
rise over run

4 types of slopes
 positive
 negative
 zero ()
 undefined slop (l)

intercept (x and yintercept)
 a point where a line intersects a coordinate axis
 xintercept is the point on the line at which y=0
 yintercept is the point on the line at which x=0
 to find xintercept, plug in 0 for y
 to find yintercept, plug in 0 for x

slopeintercept equation
y = mx + b

horizontal lines
y = some number

vertical lines
x = some number

finding distance between 2 points
use pythagorean theorem

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

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

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

Unspecified number  picking numbers
(e.g. If the length of the side of a cube decreases by onehalf, 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 onehalf)
 volume = 2 x 2 x 2 = 8
 if cube decreases by onehalf, its new length is 2  .5 (2) = 1 unit
 its new volume = 1 x 1 x 1 = 1
 Percent decreases = change/original
 (81)/8 = 7/8 = 0.875 or 87.5%

similar triangles (2)
any time two triangles each have a right angle and also share an additional right angle, they will be similar

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

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

key questions about parabola:
(1) how many times does the parabola touch the xaxis?
(2) If the parabola does touch the xaxis, where does it touch?
 In other words, the parabola touches the xaxis 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)
 (1) If b^2  4ac > 0, two roots > parabola crosses the xaxis twice and has two xintercepts
 (2) If b^2  4ac = 0, produce one root, parabola touches the xaxis once and has just one xintercept
 (3) If b^2  4ac < 0, square root operation cannot be performed, produces no roots, parabola never touches the xaxis (it has no xintercept)

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 (, )
 (4) Put the information together.

The Intersection of Two Lines
ex. at what point does the line represented by y4x10 intersect the line represented by 2x+3y=26
 replace y in the second equation with 4x10 and solve for x
 solve for y