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area of a triangle
(base x height)/2
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area of any parallelogram
base x height
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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
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volume
length x width x height
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parallelogram
opposite sides and opposite angles are equal
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trapezoid
one pair of opposite sides is parallel.
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rhombus
all sides are equal. opposite angles are equal.
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sum of interior angles of a polygon
(n-2) x 180 where n = the number of sides
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isosceles triangle
- if two sides are equal, their opposite angles are also equal
- 45-45-90
- leg:leg:hypotenuse
- 1:1:
 - x:x:x
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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
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pythagorean theorem
a2 + b2 = c2
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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)
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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
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30-60-90
- short leg : long leg : hypotenuse
- 1 : : 2
- x : x: 2x
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diagonal of a square
- s where s is an side of the cube
- also the face diagonal of a cube
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diagonal of a cube
s where s is an edge of the cube
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diagonal of a rectangle
find either the length and the width or one dimension and the proportion of one to the other
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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)
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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
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area of an equilateral triangle with a side length of S
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area of a right triangle
(1/2) Hypotenuse x height from hypotenuse
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radius
- any line segment that connects the center point to a point on the circle
- half the distance across a circle
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chord
any line segment that connects two points on a circle
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diameter
- any chord that passes through the center of the circle
- 2 x r
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circumference
- d
- 2r
- revolution (i.e. a full turn of a spinning wheel)
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area of a circle
r 2
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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
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perimeter of a sector
- corresponds to a slice of pizza
- formed by the arc and two radii (sum arc length and two radii)
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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
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central angle
an angle whose vertex lies at the center point of a circle
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inscribed angle
- has its vertex on the circle itself
- equal to half of the arc it intercepts
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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
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surface area of a cylinder
2 r 2 + 2 rh
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volume of a cylinder
r 2h
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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)
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exterior angles of a triangle
equal to the sum of the two non-adjacent (opposite) interior angles of the triangle
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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
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slope of a line
rise over run
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4 types of slopes
- positive
- negative
- zero (----)
- undefined slop (l)
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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
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slope-intercept equation
y = mx + b
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horizontal lines
y = some number
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vertical lines
x = some number
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finding distance between 2 points
use pythagorean theorem
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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
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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
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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
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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%
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similar triangles (2)
any time two triangles each have a right angle and also share an additional right angle, they will be similar
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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
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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
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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)
- (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)
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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.
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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
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