Class2 geometry and trigonometry

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TRIGONOMETRY
 Pythagorean triples
 The Pythagorean theorem states that the
square of the hypotenuse of a right
triangle is equal to the sum of the squares
of the other two sides. It can be written as
an equation
 𝑎2
+ 𝑏2
= 𝑐2
 Where c is the length of the hypotenuse
 a and b are the lengths of the other two
sides

Figura 1: Diagrama representativo de Ley de Moore
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Pythagorean Theorem
 Trigonometric and Inverse Trigonometric
Functions
 𝑆𝑖𝑛 𝜃 =
𝑂𝑝𝑝
𝐻𝑦𝑝
; 𝑎𝑟𝑐𝑠𝑖𝑛
𝑎
𝑐
= sin−1 𝑎
𝑐
= 𝜃
 𝐶𝑜𝑠 𝜃 =
𝐴𝑑𝑗
𝐻𝑦𝑝
; 𝑎𝑟𝑐𝑐𝑜𝑠
𝑏
𝑐
= cos−1 𝑏
𝑐
= 𝜃
 𝑇𝑎𝑛 𝜃 =
𝑂𝑝𝑝
𝐴𝑑𝑗
; 𝑎𝑟𝑐𝑡𝑎𝑛
𝑎
𝑏
= tan−1 𝑎
𝑏
= 𝜃

523/10/2018 Taller 2018
GENERATING TRIPLES
 a = 2mn, b = n2- m2, c = n2+ m2
 If mn is odd (m and n are of opposite
parity) and m and n are coprime (no
common factors) then the triple (a , b, c)
is primitive, that is a, b, and c have no
common factors

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UNIT CIRCLE: TRIG RATIOS FOR ANGLES GREATER
THAN 90°
 Counterclockwise =Positive angles
 Clockwise=Negative angles
 Very large angles.- are those greater than
a complete radius rotation or 360°
 Solve these equations for theta where 0° <
𝜃 < 360°. (is greater than 0 degrees and
less than 360°) State the Quadrant
number for each solution (answer to 2
d.p.)

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FIRST QUADRANT REDUCTION
We know that Sin x and Cos x are the periodical
functions with period 2pi.
Therefore, for any t=2npi + x, Sin t= Sin(2npi + x)=Sin x
and Cost =Cos(2npi + x)=cosx

923/10/2018 Taller 2018
 For Calculus -> always use radians, not
degrees
 Know the unit circle:

1023/10/2018 Taller 2018
TRIANGULATION-COSINE AND SINE
RULES
In many situations, certain geographical or
topographical features are not accesible to a survey. To
find important locations or features, triangulation is
used. This technique requires the coordination of bearing
from two known locations, for example, fire spotting
towers, to a third inaccesible location and the fire.
1. Triangulation should be used when:
 The distance between two locations is given and
 The direction from each of these two locations to the
third inaccesible location is known
2. For triangulation:
 The sine rule is used to find distances from the known
locations to the inaccesible one
 The cosine rule may be used occasionally for locating a
fourth inaccesible location

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TRIGONOMETRIC EQUATIONS
Solve for x:
𝑆𝑖𝑛𝑥 =
1
2
We know that the 𝑆𝑖𝑛
𝜋
6
=
1
2
therefore
𝑆𝑖𝑛𝑥 = 𝑆𝑖𝑛
𝜋
6
𝑥 =
𝜋
6
+2k 𝜋 or 𝑥 =
5𝜋
6
+2k 𝜋
Remember:
Sin and Cos have Period = 2 𝜋,
Whereas Tan and Cot have Period = 𝜋

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Solve for x:
𝑇𝑎𝑛𝑥 = ±
3
3
We know that the 𝑇𝑎𝑛
𝜋
6
=
3
3
or 𝑇𝑎𝑛 −
𝜋
6
= −
3
3
𝑇𝑎𝑛𝑥 = 𝑇𝑎𝑛
𝜋
6
𝑇𝑎𝑛𝑥 = 𝑇𝑎𝑛(−
𝜋
6
)
𝑥 =
𝜋
6
+ k 𝜋 and 𝑥 = −
𝜋
6
+ k 𝜋

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Solve for X:

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https://www.teachoo.com/7253/2244/Law-of-Cosine-(Cosine-
Law)/category/Sine-and-Cosine-Formula/
COSINE LAW
EXAMPLE:

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https://www.slideshare.net/vhughes5/sine-and-cosine-rule-36050547

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 A boat(B) is spotted by 2 lighthouses (A
and C), which are 17.3 km apart. The
angles measured by each lighthouse are
shown in the figure at right. Determine the
distance from each lighthouse to the boat.

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A helicopter has an altitude of 500 m and sees two cars
below. From the helicopter’s position, car A has a bearing
of 230 and car B a bearing of 120, and the angles of
depression to the cars are 38° and 55° respectively. How
far apart are the cars?

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In the rectangular coordiante system below, the are of triangular
región PQR is
A)12.5 B)14 C)10 2 D)16 E)25

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The Inverse Trigonometric
Functions
 Using the variables θ and a and view
Cos(θ)=a as a function about the unit
circle, we have the definition of Cos(θ) = a
= x coordinate of the end point (a,b) on
the unit circle defined by the arc of length
θrad.
 The cosine inverse:
 cos−1
𝑎 = 0 (−1 ≤ 𝑎 ≤ 1)

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In the rectangular coordinate system shown above, points O,P, and Q represent the
sites of three proposed housing developments. If a fire station can be built at any
point in the coordinate system, at which point would it be equidistant from all three
developments?
A)(3,1) B)(1,3) C)(3,2) D)(2,2) E)(2,3)
(Source GMAT 2018 test guide)

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In the coordinate plane, line k passes through the origin and has slope 2. If points
(3,y) and (x,4) are on line k, then x+y=
A)3.5 B)7 C)8 D)10 E)14
Guadalupe owns 2 rectangular tracts of land. One is 300 m by 500 m and the
other is 250 m by 630m. The combined área of these 2 tracts is how many
square meters?
A)3,360 B)307,500 C)621,500 D)704,000
E)2,816000

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GEOMETRY
CIRCLES
 A circle is a shape with all points having
the same distance from the center.
 It is a set of points that are all the same
distance from a central point.
 RADIUS
 DIAMETER
A chord of a circle is a line segment that has its endpoints on the
circle. A chord that passes through the center of the circle is a
diameter of the circle. A radius of a circle is a segment from the
center of the circle to a point on the circle. The words “diameter” and
“radius” are also used to refer to the lengths of these segments.

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The circumference of a circle is the distance around the circle.
22
7
= 3.1415 … = 𝜋
The área of a circle of radius r is equal to 𝜋𝑟2
A chord of a circle is a line segment that has its endpoints on the
circle. A chord that passes throigh the center of the circle is a
diameter of the circle. A radius of a circle is a segment from the
center of the circle to a point on the circle. The words “diameter” and
“radius” are also used to refer to the lengths of theses segments.

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PERPENDICULAR LINES
An angle that has a measure of 90° is a right angle. If two lines intersect at right
angles, the lines are perpendicular. For example:
𝑙1 𝑎𝑛𝑑 𝑙2 above are perpendicular, denoted by 𝑙1 𝑙2

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The number of degrees of arc in a circle (or the number of degrees in
a complete revolution) is 360.

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A line that has exactly one point in common with a circle is said to be
………… to the circle, and that common point is called the point of …………… A
radius or diameter with an endpoint at the point of tangency is
perpendicular to the tangent line, and, conversely, a line that is
perpendicular to a radius or diameter at one of its endpoints is tangent to
the circle at that endpoint.

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Two or more circles with the same center are called concentric circles,
as shown in the figure below.
A polygon is inscribed in a circle if all its vértices lie on the circle, or equivalently,
the circle is circumscribed about the polygon. Triangles RST and XYZ below are
inscribed in the circles with center O and W, respectively.

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If each vertex of a polygon lies on a circle, then the polygon is inscribed
in the circle and the circle is circumscribed about the polygon. If each
side of a polygon is tangent to a circle, then the polygon is
circumscribed about the circle and the circle is inscribed in the polygon.

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 CIRCUMFERENCE is a measure of the
distance around the circle. The
circumference is essencially the perimeter
of a circle.

𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒
𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟
= 𝜋
 ARC LENGTH, RADIUS, AND RADIUS.- Talk
about how Erastosthenes calculated the
Earth’s radius

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 TRIANGLE
 SUM OF ANY TWO SIDES > THIRD SIDE
 SUBTRACTION OF TWO SIDES < THRID
SIDE
 SUM OF THE THREE ANGLES = 180°
 SAME SIDES=SAME ANGLES, AND
VICEVERSA
 PERIMETER=SUM OF SIDES
FOR US TO REMEMBER:

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TRIANGLE INEQUALITY THEOREM
The sum of the lengths of any two sides of a triangle is greater than the
length of the third side.
https://www.onlinemathlearning.com/triangle-inequality.html

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In pentagon PQRST, PQ=3, QR=2, RS=4, and ST=5. Which of the lengths 5,10,
and 15 could be the value of PT?
(A) 5 only
(B) 15 only
(C) 5 and 10 only
(D) 10 and 15 only
(E) 5, 10 and 15
Note: Not drawn to scale

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The área of a triangle is equal to:
𝑡ℎ𝑒 𝑙𝑒𝑛𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 𝑥(𝑡ℎ𝑒 𝑙𝑒𝑛𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒)
2

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The área of a paralelogram is equal to
𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 𝑥(𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒)
A parallelogram with the right angles is a rectangle, and a rectangle with all
sides of equal length is a …………..

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In the figure above, triangle ABC is equilateral, and point P is equidistant from
vértices A, B, and C. If triangle ABC is rotated clockwise about point P, what is the
mínimum number of degrees the triangle must be rotated so that point B will be in
the position where point A is now?
A)60 B)120 C)180 D)240 E)270
Source GMAT Test Guide 2018

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POLYGONS (Convex)
A polygon is a closed plane figure formed by three or more line segments, called
the sides of the polygon. Each side intersects exactly two other sides at their
endpoints. The points of intersection of the sides are vértices. The term
“polygon” will be used to mean a convex polygon, that is, a polygon in which
each interior angle has a measure of less than 180°
The following figures are polygons:
The following figures are not polygons:

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The figure shown above consists of a shaded 9-sided polygon and 9 unshaded
isósceles triangles. For each isósceles triangle, the longest side is a side of the
shaded polygon, and the two sides of equal length are extensions of the two
adjacent sides of the shaded polygon. What is the value of a?
A)100 B)105 C)110 D)115 E)120

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In the figure shown, if the are of the shaded región is 3 times the área of the smaller
circular región, then the circumference of the larger circle is how many times the
circumference of the smaller circle?
A)4 B)3 C)2 D) 3 E) 2

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The parallelogram shown has four sides of equal length. What is the ratio of the length
of the shorter diagonal to the length of the longer diagonal?
A)1/2 B)1/ 2 C) 1/2 2 D) 1/ 3 E) 1/2 3

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A window is in the shape of a regular hexagon with each side of length 80
centimeters. If a diagonal through the center of the hexagon is w centimeters long,
the w=
A)80 B)120 C)150 D)160 E)240

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In the figure shown, PQRSTU is a regular polygon with the sides of length x. What is
the perimeter of triangle PRT in terms of x?
A)
𝑥 3
2
B)𝑥 3 C)
3𝑥 3
2
D)3𝑥 3 E)4𝑥 3

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In an electric circuit, two resistors with resistances x and y are connected in parallel. In
this case, if r is the combined resistance of these two resistors, then the reciprocal of r is
equal to the sum of the reciprocals of x and y. What is r in terms of x and y?
A)xy B)x+y C)1/(x+y) D)xy/(x+y) E)(x+y)/xy

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Viewed from the outside inward, the figure below depicts a square-circle-
square-circle, each enclosed with the other. If the área of square ABCD is 2
square units, then which of the following expresses the are of the darkened
corners of square EFGH?
Find the Area of the Shaded Region
A)2-π/4 B)2-π/2 C)1-π/4 D)1/2-π/8 E)1-π/2

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QUADRILATERALS
 A quadrilateral is any figure with 4 sides.
Three common variations are described
below. Before getting to them, realice that
you can always cut up any quadrilateral
into two triangles by slicing across the
middle to connect opposite corners.

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A polygon with four sides is a quadrilateral
The área of a parallelogram is equal to
𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 𝑥(𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒)

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Parallelogram OPQR lies in the xy-plane, as shown in the coordinates of point P are
(2,4) and the coordinates of point Q are (8,6) What are the coordinates of point R?
A)(3,2) B)(3,3) C)(4,4) D)(5,2) E)(6,2)

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A parallelogram with right angles is a ………., and a rectangle with all sides
of equal length is a ………….
A quadrilateral with two sides that are parallel, as shown above, is a
………………. The área of a trapezoid PQRS may be calculated as follows:
1
2
𝑡ℎ𝑒 𝑠𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒𝑠 𝑡ℎ𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 =
1
2
𝑄𝑅 𝑃𝑆 8 = 112

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DISCRETE QUESTION: HARD
Quantitative Comparison
A Quantity A is greater
B Quantity B is greater
C The two quantities are equal
D The relationship cannot be determined from the information given
A symbol that appears more than once in a question has the same meaning
throughout the question
Source: GRE 2011 test

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If one side of an inscribed triangle is a diameter of the circle, as in triangle
XYZ above, then the triangle is a right triangle. Conversely, i fan inscribed
triangle is a right triangle, then one of its sides is a diameter of the circle.
A polygon is circumscribed about a circle if each side of the polygon is
tangent to the circle, or equivalently, the circle is inscribed in the polygon.
In the figure below, quadrilateral ABCD is circumscribed about the circle
with center O.

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In the figure shown, the triangle is inscribed in the semicircle. If the length
of line segment AB is 8 and the length of line segment BC is 6, what is the
length of arc ABC?
A)15π B)12π C)10π D)7π
E)5π

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TRIANGLE ANGLE BISECTOR THEOREM
An angle bisector of an angle of a triangle divides the opposite side
in two segments that are proportional to the other two sides of the
triangle.
Example: Find the value of x

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AREA OF TRIANGLE IN TERMS OF CIRCUMRADIUS
𝐴 =
𝑎𝑏𝑐
4𝑅

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Brahmagupta’s Formula: Area of cyclic quadrilateral
If a cyclic cuadrilateral ABCD has sides of lengths a, b, c and d then:
Heron’s formular is a special … of this formula, when one of the
sides equals zero.

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 RECTANGLES = PARALLELOGRAM + 4
RIGHT ANGLES
 SQUARES=RECTANGLE + 4 EQUAL
SIDES
 GEOMETRY WORD PROBLEMS WITH
PICTURES
SUMMARY

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COORDINATE PLANE-POSITION IS A
PAIR OF NUMBERS
 PLOT A RELATIONSHIP:GIVE ME AN X, I’LL TELL
YOU A Y
 If you plug a number in for x in any of
these equations, you can calculate a value
for y.
x Y = 2x + 1
-2
-1
0
1
2

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LINES IN THE PLANE: USE SLOPE AND Y-
INTERCEPT TO PLOT
 𝑆𝑙𝑜𝑝𝑒 =
𝑅𝑖𝑠𝑒
𝑅𝑢𝑛
=
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥

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3D SHAPES

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The other relationship between these values is
given by Euler's formula:
𝑉 − 𝐸 + 𝐹 = 2
A convex polyhedron is a Platonic solid if and only if all its faces
are congruent convex regular polygons, none of its faces intersect except at their
edges, and the same number of faces meet at each of its vertices.
PLATONIC SOLIDS

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In the rectangular solid above, TU=3, UV=4, and VR=2. What is the área of
the shaded rectangular región?
A list of numbers has a mean of 8 and a estándar deviation of 2.5. If x is a
number in the list that is 2 standard deviations above the mean, what is the
value of x?
Source: GRE

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A closed cylindrical tank contains 36π cubic feet of wáter and is filled
to half its capacity. When the tank is placed upright on its circular
base on level ground, the height of the wáter in the tank is 4 feet.
When the tank is placed on its side on level ground, what is the
height, in feet, of the surface of the wáter above the ground? Gmat
2018
A)2 B)3 C)4 D)6 E)9

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RECTANGULAR SOLIDS AND CYLINDERS
A rectangular solid is three-dimensional figure formed by 6 rectangular surfaces,
as shown below. Each rectangular surface is a face. Each solid’s dotted line
segment is an edge, and each point at which the edges meet is a vertex. A
rectangular solid has 6 faces, 12 edges, and 8 vertices. Opposite faces are parallel
rectangle that have the same dimensions. A rectangular solid in which all edges
are of equal length is a cube.
The surface are of a rectangular solid is equal to the sum of the áreas of all
the faces. The volumen is equal to
𝑙𝑒𝑛𝑔𝑡ℎ 𝑥 𝑤𝑖𝑑𝑡ℎ 𝑥 ℎ𝑒𝑖𝑔ℎ𝑡 ;
In other words, 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 𝑥(ℎ𝑒𝑖𝑔ℎ𝑡)

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The volumen of a cylinder is equal to 𝜋𝑟2ℎ. That is,
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 𝑥(ℎ𝑒𝑖𝑔ℎ𝑡)

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GMAT 2016 Test Guide
A right circular cone is inscribed in a hemisphere so that the base of
the cone coincides with the base of the hemisphere. What is the ratio
of the height of the cone to the radius of the hemisphere?
(A) 3: 1 (A)1: 1 (A)1/2: 1
(A) 2: 1 (A)2: 1
Geometry Volume
As the diagram below shows, the height of the cone will be the radius
of the hemisphere, so the ratio is 1:1

65
[1]The official Guide for GMAT review 2016
Graduate management admission council mba.com
[2]GMAT Official Guide 2018
Graduate Management Admission Council mba.com
[3]Manhattan GMAT Foundations of GMAT Math 5th Edition-GMAT Strategy Guide Arithmetic and Algebra
Dan Gonzales
[3]The oficial Guide to GRE revised general test 2nd Edition
Mc Graw-Hill Education product www.mhprofessional.com
[4] The Platonic Solids
[Online] Available:
http://www.technologyuk.net/mathematics/geometry/platonic-solids.shtml
[5] Geometry: Triangle Inequality and Angle-Side Relationship
Volnei A. Pedroni
https://www.onlinemathlearning.com/triangle-inequality.html
[6]Law of Cosine
[Online] Available:
https://www.teachoo.com/7253/2244/Law-of-Cosine-(Cosine-Law)/category/Sine-and-Cosine-Formula/
[7] GMAT- Find the área of the shaded región
https://www.youtube.com/watch?v=p_fICxSBbmA
[8] GMAT strategy: the power of using common sense by Thomas Paine
https://www.youtube.com/watch?v=xBKZ-TsGkKw
23/10/2018 Taller 2018

6623/10/2018 Taller 2018
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Class2 geometry and trigonometry

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Class2 geometry and trigonometry