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MATH ONLY > GEOMETRY > Flashcards

GEOMETRY Flashcards

1
Q

3 Principles of Geometry Strategy

A
  1. If they don’t tell you, don’t assume - diagram lie!
  2. If they give you a piece of info, use it!
  3. Know the rules + formulas
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2
Q

On the coordinate plane
x = ?
y = ?
o = ?

A

x = the line and numbers moving right to left

Y = the line and numbers moving down to up

O = the origin point, (0,0) where the lines meet.

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3
Q

Label the Quadrants of the Plane

A

(x,y)
I = + +
II = - +
III = - -
IV = + -

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4
Q

What is the Slope of a Line?

A

The measure of aline’s incline on the the coordinate plane. The formula is rise / run or y2 - y1 / x2 - x1

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5
Q

In the formula y = mx + b what do m and b represent?

A

m = the slope of the line
b = the y intercept or the point where the line cross the y axis

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6
Q

What is the Slope Intercept Equation?

A

y = mx + b

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7
Q

A line is expressed as y = -5/6x + 5, which is greater, the x or y intercept?

A

the Y intercept is give as 5. You can calculate the X intercept by setting y to 0 and solving for x.

0 = -5/6x + 5
5/6x = 5
5x = 30
x = 6

The X intercept > The Y intercept

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8
Q

Intersecting Lines Knowns?

A
  • The sum of all resulting angles = 360
  • Intersecting lines angles that for a line = 180 degrees
  • When two lines intersect angles found opposite to each other are equal
  • All acute angles are less than 90 degrees
  • All obtuse angles are greater than 90 degrees
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9
Q

Parallel Lines cut by a Transversal Line Knowns?

A

All of Intersecting Lines Knowns +

II - in writing means that lines are parallel

The angles created by the cut will mirror each other. Basically it creates to set of angles that are clones with each other.

Look out for how they disguise this, Technically a Z is a set of parallel lines cut by a Transversal line.

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10
Q

Formula for the Sum of the Interior Angles of a Polygon?

A

n = number of sides

(n-2) * 180 = total degree count

E.G. triangles and squares

Triangle: (3-2) * 180 = 180

Square: (4-2) * 180 = 360

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11
Q

A Quadrilateral Is?

A

Any shape with 4 sides.

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12
Q

A Parallelogram is?

A

Any four sided figure in which the opposite side are parallel and equal.

As a result
- Opposite angles are also equal
- Adjacent angles = 180
- A diagonal through the transverse middle = two equal triangles
- Perimeter = 2 (Side A + Side B)

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13
Q

How do you find the area of a Parallelogram?

A

Base * Height but the height of the triangle is taken from a right angle line made from the base. The height is not the side.

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14
Q

How do you find the area of a Trapezoid?

A

Area of Trap = (Base1 + Base2) * (Height) / 2

The height is taken from a right angle line made from the base. The height is not the side.

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15
Q

How do you find the volume of a cube?

A

Volume of a cube = Length * Width * Height or Edge Length aka E (because L, W, H are all equal in a cube) to the third power or E^3

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16
Q

How do you find the surface area of a cube?

A

Sum of the area of the faces = surface area

or

e = edge of cube

so surface area = 6e^2

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17
Q

Formula for the area of a Triangle?

A

Base * Height * 1/2

For equilateral : s^2 * sqrt 3 / 4

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18
Q

The internal angles of a triangle must equal?

A

180 degrees

If you know any two angles of a triangle you know the third.

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19
Q

The exterior angle of a triangle can be derived by?

A

If you confirm that the third angle lies on a line then the sum of the two non adjacent interior angles = the exterior angle or the 180 - the adjacent angle = the exterior angle.

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20
Q

In a Triangle:

The longest side is opposite?

The shortest side is opposite?

A

Longest side is opposite the greatest angle

Shortest side is opposite the smallest angle

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21
Q

In a Triangle:

Two Equal Sides means

Two Equal Angle menas

A

Two equal sides means the opposite angles are equal and two equal angles means the sides opposite are equal.

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22
Q

The Pythagorean applies only to?

A

Right Triangles

a^2 + b^2 = c^2

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23
Q

What are the common Pythagorean Triples on the GMAT for Right Triangles

A

8-15-17
5-12-13 –> also comes as 10-24-26
3-4-5 –> Also comes as multiple versions that multiply the base numbers by 2, 3 or 4

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24
Q

What is the ratio of the lengths of the sides of a 30, 60, 90 degree triangle?

A

x : 1.7x : 2x

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25
Q

Triangles:

The sum of any two sides must?

The difference of any two sides must?

A

The sum of any two sides must be greater than the third side.

The difference of any two sides must be less than the third side.

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26
Q

What is a Diameter of the circle

A

A line that bisects the circle across it’s center point or 2 * the radius. d = 2r

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27
Q

What is this?

A

A circumference or the perimeter of the circle. C = Pi * d or Pi * 2r

Pi = Circumference / Diameter

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28
Q

Formula for Area of a Circle?

A

A = Pi * r^2

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29
Q

Pieces of a circle are called?

A

Sectors

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30
Q

How do you find the fraction a sector of a circle represents?

A

Central Angle / 360 = Sector Fraction

Central angle is taken from the middle of the circle.

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31
Q

An inscribe angle is = to?

A

Inscribed Angle = 1/2 * Corresponding Central Angle

If one side of the inscribed angle is the diameter it can be a inscribed right triangle.

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32
Q

Arc Length of a Sector of a Circle = ?

A

Circumference * Sector Fraction = Arc Length

33
Q

Sector Area of a Circle = ?

A

Sector Area = Sector Fraction * Full Area of Circle

34
Q

At what angle does a line tangent to a circle meet that circle’s radius at the point of tangency?

A
35
Q

Formula for Volume of a Cylinder?

A

Volume = Pi * r^2 * h

36
Q

Formula for Surface Area of a Cylinder

A

Lateral Surface Area aka Just the tube = 2 * Pi * r * h

Total area = (2 * Pi * r^2) + (2 * Pi * r * h)

37
Q

In a circle a Chord is any line that? And is always …. than the diameter.

A

In a circle a Chord is any line that is not the radius And is always SHORTER than the diameter.

38
Q

What is this?

A

A radius or a line that crosses the circle from its center to its edge.

39
Q

Coordinate Geometry - Every Line…

A

Has its own unique equation! All of them, every single variation to possibly exist.

40
Q

Coordinate Geometry - For any give line all points on the line…

A

Will satisfy the equation of the line. That’s basically an infinite amount of points.

41
Q

Coordinate Geometry - Any linear equation that as X^1 + Y^1 = Z with no multiplication or division must…

A

Be the equation of some line in the plane! That’s why they are called linear equations.

42
Q

A line passes through (A, 30) with a y intercept of (0,1) and a slope of 1/2, What is A?

A

Take the slope intercept equation - Y = MX + B and plug in what we know

30 = 1/2A + 1

Then solve for A!

A = 58

43
Q

Any point I can find the Y intercept by….

A

Plugging in a Y coordinate into Y and an X coordinate into X and the slope into M - then I solve for B!

44
Q

I can find the X intercept by…

A

Solving for the Y intercept and slope. I can then set Y to 0 and solve for X!

45
Q

Area of an equilateral Triangle…

A

Sqrt3/4 * (side^2)

46
Q

Y = 3 is a …

A

Horizontal line. Y = any integer with no X intercept represented = a horizontal line. The integer represents the Y coordinate for all points on the line.

47
Q

X = 4

A

A vertical line. x = any integer with no y intercept represented = a vertical line. The integer represents the x coordinate for all points on the line. Which is really just one point.

48
Q

When I draw out a table to create coordinates for a line I need…

A

A starting point and the slope. The table looks like follows:

49
Q

Parallel lines have slopes that are…

A

equal

50
Q

Perpendicular lines have slopes that are

A

Negative reciprocals of the opposite line. Always add a negative to the reciprocal. It will render a positive where appropriate. EG -2/3 negative reciprocal is 3/2 bcz you flip and add neg which cancels out the neg of the original slope.

51
Q

When I see line equations and graphs I think about…

A

Using Y or X intercepts to eliminate possibilities. I find them by plugging in 0 into the opposite variable. So the equation for the X intercept Y = 0 and vice versa.

52
Q

Y = X has a slope of….
Y = -X has a slope of….

A

Y = X has a slope of 1
Y = -X has a slope of -1

53
Q

Coordinates of Reflections over Y = X are ….

and are equidistant from ….

A

Flip flops of X,Y coordinates of their mirror image. So 1,2 is 2,1

These points are equidistant from any point on their mirror line

54
Q

All points above y = x are…

All points below y = x are…

A

Above : Y > X

Below : X > Y

55
Q

Reflections over x axis are …

Reflections over y axis are …

Reflections over Y = X axis you …

Reflections over Y = -X axis you …

A

1 . Same X opposite +/- Y

  1. Same y opposite +/- X
  2. Switch X, Y coordinates
  3. Switch X, Y coordinates and make their signs opposite
56
Q

Mirror line is …

A

Always the perpendicular bisector of the segment and its reflected image

57
Q

Graph of a Quadratic is …

A

A parabola

58
Q

A Vertex of a parabola is …

A

Its lowest or highest point

59
Q

In the equation of a parabola

ax^2 + bx + c

what happens when:

  1. a > 0
  2. a < 0
  3. |a| > 1
  4. |a | < 1
A

1 . parabola is a U opening up

  1. Parabola is a reverse U opening down
  2. Parabola is skinny
  3. Parabola is wide
60
Q

Parabolas can have how many x intercepts?

A

0 , 1 or 2

61
Q

Equation of a circle…

What do a, b stand for?

A

(x-a)^2 + (y-b)^2 = r^2

r = radius
a = x coordinate of center of circle
b = y coordinate of center of circle

62
Q

Similar Shapes are…

A

Shapes with the same ratios & angles but at different proportions. By establishing similarity several inferences can be drawn.

For example triangles with identical angles are similar and information from one can be used to find info on the other.

63
Q

Scale Factor in Similar Triangles =

A

Side of the Bigger Triangle / Corresponding Side of the Little Triangle = Scale Factor

64
Q

Area of a bigger similar triangle derived from knowing sides of smaller triangle…

A

1/2 * B * H * X^2 = Area of big triangle

BH = base and height of small triangle
X = Scale Factor

65
Q

An Isosceles Right Triangle has a Hypotenuse with what proportion to its other legs?

A

An Isosceles Hypotenuse to Other Legs has a ratio of SqRt 2 to 1.

This ratio will also hold true for the diagonal of all squares as they are simply two IR triangles put together.

66
Q

An equilateral triangle is what two other triangles put together?

A

Two 30:60:90 triangles back to back.

67
Q

Rhombus is…

A

All sides are equal and all diagonals are perpendicular

68
Q

Rectangles is…

A

Have 90 degree angles x4 and diagonals are congruent.

69
Q

Square is…

A

Properties of a Rectangle, Rhombus and Parallelogram put together.

70
Q

Trapezoid is…

A

One pair of parallel sides (symmetrical trapezoids have equal lines)

71
Q

When I have to find the length of any slanted line I think of…

A

Using the pythagorean theorem aka how do I build a right triangle with the line I need to find as one of the sides.

72
Q

When Something is a Regular Polygon it means…

A

That actually it’s a special polygon that has all equal sides and equal angles. Like a Square or Equilateral Triangle.

73
Q

A triangle inscribed in a circle with a hypotenuse as the diameter is…

A

An isosceles right angle triangle, any inscribed angle with the diameter as its chord is 90 degrees

74
Q

Two inscribed angles with the same arc or chord are…

A

equal to each other

75
Q

Arc Length / 2 * Pi * R =

A

Arc Length / 2 * Pi * R = Central Angle / 360

  • You can use this as a proportion to find one of these missing factors
76
Q

Area of Sector / Pi * R^2 =

A

Arc Length / 2 * Pi * R = Angle / 360

  • You can use this as a proportion to find one of these missing factors
77
Q

3D object, how do you find the space diagonal of a rectangle?

A

Diagonal^2 = L^2 + W^2 + H^2

78
Q

3D object, how do you find the space diagonal of a square?

A

Diagonal = SqRt 3 * Side

79
Q

Area increases between similar shapes by the scale factor k being squared so…

A

To find the area of the larger object I times the area of the smaller object by k^2

MATH ONLY (12 decks)

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