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Summary Manhattan GMAT - Geometry
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Geometry1 Polygons
GMAT works with the following polygons:
Three-sided shapes (triangles – driehoeken)
Four-sided shapes (quadrilaterals – vierhoeken)
Other polygons with n side (where n is five or more)
Different types of quadrilaterals: Thombus
All sides are equal.
Parallelogram Rectangle Square
Opposite sides and All angles are 90, and All angles
opposite angles are equal. opposite sides are equal. are 90. All
sides are
equal.
Rectangles and rhombuses are special types of parallelograms. And note that a
Square is a special type of parallelogram that is both a rectangle and a rhombus.
ne pair of opposite sides is parallel.
rapezoid
The sum of the interior angles of a polygon follows a specific pattern that depends
on n, the number has. This sum is always 180 times 2 less than n.
Polygon # of sides Sum of interior angles
Triangle 3 180
Quadrilateral 4 360
Pentagon - vijfhoek 5 540
Hexagon - zeshoek 6 720
This pattern can be expresses with the following formula:
(2-n) x 180 = sum of interior angles of a polygon
Perimeter – omtrek
The sum of the lengths of all sides.
Area – oppervlakte
1. Area of a triangle
base x height / 2
2. Area of a rectangle
length x width
3. Area of a trapezoid
(base1 + base2) x height / 2
4. Area of any parallelogram
base x height
5. Area of a rhombus
diagonal1 + diagonal
Surface area – oppervlakte van alle vlakken
GMAT test two shapes: the rectangular solid and the cube.
Surface area = the sum of all the faces
Volume
Volume = length x width x height
, 2 Triangles & diagonals
The most important property of a right triangle is the unique relationship of the three sides.
Given the lenths of any two of the side of a right triangle, you can determine the length of the
third side using the Pythagorean theorem. There are two types of right triangles:
1. 30 – 60 – 90
2. 45 – 45 – 90
For the second you only need to know the length of one side.
The sum of the three angles of a triangle equals 180.
If two sides of a triangle are equal, their opposite angles are also equal.
The sum of any two sides of a triangle must be greater than the third side (thus,
also not equal to the third side).
If you are given two sides of a triangle, the length of the third side must lie between
the difference and the sum of the two given sides.
The Pythagorean theorem – stelling van Pythagoras
A right triangle is a triangle with one right angle (90). Every right triangle is composed of two
legs and a hypotenuse (schuinde zijde). The hypotenuse is the side opposite the right angle
and is often assigned the letter c. the two legs which form the right angle are often called a
and b.
Pythagorean theorem: a2 + b2 = c2
The sum of the square of the two legs of a right triangle (a and b) is equal to the square of
the hypotenuse of that triangle (c).
You should know the following Pythagorean triples form memory. They are often asked:
Common combinations Key multiplies
3–4–5 6 – 8 – 10
The most popular of all right triangles 9 – 12 – 15
32 + 42 = 52 (9+16=25) 12 – 16 -20
5 – 12 -13
Also quite popular on the GMAT 10 – 24 - 26
52 + 122 = 132 (25+144=169
8 – 15 - 17
This one appear less frequently None
82 + 152 = 172 (64+225=289
Altijd eerst kijken of de driehoek wel een rechte driehoek is!
Isoscles triangels – 45 – 45 – 90 triangle
The lenght of the legs of every 45-45-90 triangle have a specific ratio, which you must
memorize:
45 45 90
Leg leg hypotenuse
1 : 1 : √2
X x : x√2
Equilateral triangles and the 30 – 60 – 90 triangle
An equilateral triangle is one in which all three sides are equal.
30 60 90
Short leg long leg hypotenuse
1 : √3 : 2
X : x√3 : 2x
The height of an equilateral triangle Is the same as the long leg.
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