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GMAT > Geometry > Flashcards

Geometry Flashcards

1
Q

vertex (vertices)

A

a corner of a triangle

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2
Q

base

A

any side of a triangle

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3
Q

altitude

A

the line from the vertex and perpendicular to its opposite base

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4
Q

orthocenter

A

the joint point of three altitudes

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5
Q

median

A

the line from vertex to midpoint of its opposite base

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6
Q

centroid

A

joint point of three medians

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7
Q

when do we have 6 smaller triangles of equal area?

A

when we have the centroid

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8
Q

Ratio of vertex to centroid

A

2/3

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9
Q

Ratio of centroid to midpoint of triangle base

A

1/3

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10
Q

What is one of the formula to calculate area of triangle?

A

(P. r )/ 2
P = Perimeter of triangle
r = the radius of inscribed circle

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11
Q

What is one of the formula to calculate area of triangle?

A

(h.b) /2
h = altitude of triangle
b = base of triangle

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12
Q

What is one of the formula to calculate area of triangle?

A

(abc)/ 4R
a,b,c = bases of triangle
R = radius of circumscribed triangle

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13
Q

What is one of the formula to calculate area of triangle?

A

√s(s-a)(s-b)(s-c)

s = semi-perimeter of triangle

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14
Q

Two triangles are similar if

A
  • their corresponding sides are in proportion

- square root of their area ratio equal to ratio of their bases

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15
Q

T or F. with corresponding sides in ratio of x/y, their volume will be in a ratio x^3 / y^3 for two similar solids

A

T

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16
Q

What does the two similar triangles theory extend?

A

to any 2-D figures

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17
Q

T or F. The shortest side is always opposite the largest interior triangle

A

False.

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18
Q

T or F. If two sides are equal, their opposite angles are equal

A

True.

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19
Q

T or F. The longest side is always opposite the largest interior triangle

A

True.

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20
Q

angle bisector

A

the line (from vertex) that divides a triangle angle into two equal angles.

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21
Q

incenter

A

the joint point of three angle bisectors

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22
Q

What is the incenter regarded as?

A

the center of inscribed circle in a triangle

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23
Q

a triangle is isosceles if:

A
  • the triangle with two sides are equal in length

- the two angles are equal to each other

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24
Q

Formula to find the unknown base given the legs and altitude (in isosceles triangle).
How does this formula derive from?

A

B = 2. √(L^2 - A^2)

Pythagorean theorem

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25
Q

a triangle is equilateral if:

A
  • all sides have same length, or

- all angles have 60 degrees

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26
Q

What is the formula to calculate area of equilateral triangle?

A 
A = a^2 . (√3 / 4)
a = length of the side
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27
Q

What is the formula to calculate the altitude of equilateral triangle?

A

h = a. (√3 / 2)

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28
Q

What is the formula to calculate the radius of circumscribed circle of equilateral triangle?

A

R = a. (√3 / 3)

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29
Q

What is the formula to calculate the radius of inscribed circle of equilateral triangle?

A

r = a. (√3 / 6)

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30
Q

For any point P within an equilateral triangle,

A

the sum of the perpendiculars to the three sides is equal to the altitude of the triangle.


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31
Q

T or F. Isosceles right triangle has the largest area in any right triangle

A

True

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32
Q

What can we say about the distance from the midpoint of hypotenuse to each vertex in right triangle?

A

they are equal to each other

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33
Q

What are the ratio of sides in right triangle?

A

3:4:5, 5:12:13 or 8:15:17

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34
Q

What are the ratio of sides in isosceles right triangle?

A

1:1 : √2

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35
Q

What are the ratio of sides in 30-60-90 right triangle?

A

1: √3 : 2

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36
Q

What is one of the formula for the radius of the circle inscribed in a right triangle?

A

r = product of two sides (not the hypotenuse) / sum of three sides

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37
Q

What is one of the formula for the radius of the circle inscribed in a right triangle?

A

r = (sum of two sides - hypotenuse)/ 2

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38
Q

The sum of the interior angle of a polygon with n sides

A

(n-2) . 180

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39
Q

The perimeter of a polygon

A

the sum of the lengths of its side

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40
Q

T or F. If the diagonals of a rhombus are equal, then it must be a square

A

T

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41
Q

What is the ratio of diagonal of square to its side?

A

1.414 or √2

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42
Q

T or F. Of all quadrilaterals, with a given perimeter, the square has the smallest area

A

F.

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43
Q

T or F. Of all quadrilateral, with a given area, the square has the minimum perimeter

A

T

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44
Q

Calculation of an area of square from knowing the diagonal

A

d^2 / 2

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45
Q

Area of circle circumscribed square

A

(pi/ 2) . a^2

46
Q

Area of circle inscribed the square

A

(pi/ 4) . a^2

47
Q

what are the characteristics of sides and angles in rhombus?

A

all side are equal

opposite angles are equal

48
Q

what is the characteristic of diagonals in rhombus?

A

their intersection forms a right angle

49
Q

Area of rhombus

A

product of two diagonals divide by 2

50
Q

Area of trapezoid

A

half of the product of height and sum of the base lengths

51
Q

Isosceles trapezoid

A

trapezoid that has equal sides that are not parallel and both angles are same

52
Q

What is the length of the median in trapezoid?

A

the average length of the bases

53
Q

What is the length of minor arc in a circle, where x is the central angle of the arc in degree?

A

(2 π r x)/ 360

54
Q

What is the area of the minor arc in a circle, where x is the central angle of the arc in degree?

A

x .π r^2/ 360

55
Q

If the angle is not at the center, but at a different point on the circle then what is the angle of that minor arc?

A 
angle = 90 L/ π r
L = length of the minor arc

angle = 1/2 of angle of the minor arc at the center

56
Q

T or F. It’s not an inscribed triangle if all of the vertices of the triangle are not points on the circle.

A

T.

57
Q

T or F. If one of the sides of an inscribed triangle is a diameter of the circle, then the triangle must be an equilateral triangle

A

F.

The triangle must be a right triangle

58
Q

If a circle circumscribes right triangle, then its radius is

A

half the length of the hypotenuse

59
Q

What does revolution means in circle?

A

one revolution = one circumference

60
Q

If a wheel spins at 3 revolutions per second then how fast a point on spinning wheel travel?

A

3 x 2. π.r (unit of length/ second)

61
Q

What is the surface area of a cube?

A

6a ^2

62
Q

Diagonal length of a cube

A

a . √3

63
Q

what is the other name for cuboid?

A

rectangular solid

64
Q

What is the surface area of a cuboid?

A

2 (ab+ bc+ ac)

65
Q

Diagonal length of a cuboid

A

√a^2 + b^2 + c^2

66
Q

Volume of cylinder

A

π.r^2. h or area of circle x height

67
Q

Outer surface area of cylinder w/o base

A

2πrh or perimeter of circle x height

68
Q

Outer surface area of cylinder w/ base

A

2πrh + 2πr^2 or

Outer surface area w/o base + area of two circles

69
Q

in a cone solid, what is a lateral height?

A

the hypotenuse of the triangle formed by the height and radius

70
Q

Volume of cone

A

a third of the product of height and area of circle (π.r^2 .h )/ 3

71
Q

Outer surface area of a cone w/o base

A

half product of circle perimeter and lateral height (πrl )

72
Q

Outer surface area of a cone w/ base

A

Outer surface area w/o base + area of circle (πrl + πr^2)

73
Q

Volume of sphere

A

(4/3).π. r^3

74
Q

Surface area of sphere

A

4πr^2

75
Q

For given two points (x1,y1) and (x2, y2) the distance between two points is

A

√(x2 - x1)^2 + (y2- y1)^2

76
Q

For given two points (x1,y1) and (x2, y2) the midpoint is

A 
x3 = (x1 + x2)/2
y3 = (y1 + y2) /2
77
Q

How do you find the slope of a line?

A

difference of y coordinate/ difference of x coordinate

78
Q

when does slope is zero happen?

A

when the line is horizontal. Thus

difference of y over difference of x is ZERO

79
Q

when is the slope undefined?

A

when the line is vertical. Thus

difference of x (numerator) is zero.

80
Q

If the slope equal to one, then what can we conclude?

A

the angle forms with the line is 45 degree

81
Q

what can we say about the slopes of two parallel line?

A

they are constant and equal to each other

82
Q

Distance between two parallel lines

A

k1 - k2 | / √(slope^2 + 1)

83
Q

T or F. the two lines are perpendicular if and only if the product of their slopes is 1

A

F.

-1

84
Q

if m is the slope of one line, what is the value of slope that perpendicular to that line

A

-1/m

85
Q

The two lines a.x + b.y + c = 0 and m.x + n.y + q = 0 are perpendicular if

A

a.m + b.n = 0

86
Q

If two lines cross each other, there is one intersected point (x,y) then at that point, the equation of line is

A

y(1) = y(2)

-> a.x + m = b.x + n

87
Q

T or F. The distance from a point to a line is shortest when the segment from that point perpendicular to the line

A

T

88
Q

The distance from a point (m, n) to a line ax + by+ c = 0 is given by the formula

A

m.a + b.n + c | / √(a^2 + b^2)

89
Q

what is the expression of a parabola?

A

It is a quadratic expression: ax^2 + bx + c

90
Q

T or F. The smaller the absolute value of a, the bigger (more open) the parabola is

A

T. since the value of y increase less quickly

91
Q

If a is positive, the parabola open

A

upward

92
Q

If a is negative, the parabola open

A

downward

93
Q

at x- intercept, the parabola expression is written as

A

ax^2 + bx + c = 0 since y =0

94
Q

at y - intercept, the parabola expression is written as

A

y = c since x = 0

95
Q

If the discriminant is positive, then parabola has

A

two intercept with x-axis or two solutions of x

96
Q

If the discriminant is negative, then parabola has

A

no intercepts with x-axis or no solutions of x

97
Q

If the discriminant is zero, then parabola has

A

one intercept with x-axis - that is a tangent point

98
Q

What is the expression of discriminant in parabola?

A

b^2 - 4ac

99
Q

What does the vertex value represent?

A

the maximum or minimum value of the quadratic function

100
Q

When is the vertex maximum?

A

when a is negative and parabola open downward

101
Q

When is the vertex minimum?

A

when a is positive and parabola open upward

102
Q

How do we calculate vertex of the parabola?

A

( -b/2a; c- b^2/4a )

103
Q

T or F. For one line (slope = m) to be perpendicular to another, the other slope is -m

A

F. it must be -1 /m

104
Q

T or F. When you are fitting the 3D objects into other 3-D objects, you can divide the volume of bigger object by volume of smaller object

A

F.

You need to know the dimension of each unit

105
Q

What do you need to know when you try to determine whether an object can fit in another one?

A

specific dimensions: length, width and height

106
Q

With given the lengths of two sides of triangle, in what conditions, the triangle has maximum area?

A

when the two sides meet at right angle.

107
Q

Given the lengths of two sides of parallelogram, in what condition will the parallelogram have the maximum area?

A

when this parallelogram is a rectangular (short side meets long side at right angle)

108
Q

In coordinate geometry, x-intercept is

A

the line intercept x-axis only and therefore y =0

109
Q

In coordinate geometry, y-intercept is

A

the line intercept y-axis only and therefore x= 0

110
Q

Area of any parallelogram

A

base x height

111
Q

Letting R = (x,y), then R will be equidistant from (−3,−3) and (1,−3) when?

A

if and only if R lies on the perpendicular bisector of the line segment with endpoints (−3,−3) and (1,−3)

112
Q

The equation of a straight line whose x and y intercepts are a and b, respectively, is:

A

x/a + y/b = 1

GMAT (16 decks)

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