## Thursday, November 19, 2009

### GMAT/CAT PREPARATION EMAIL COURSE DAY 25

3-4-5 Triangle:
If a right triangle's leg-to-leg ratio is 3:4, or if the leg-to-hypotenuse ratio is 3:5 or 4:5, it's a 3-4-5 triangle.

5-12-13 Triangle
If a right triangle's leg-to-leg ratio is 5:12, or if the leg-to-hypotenuse ratio is 5:13 or 12:13, then it's a 5-12-13 triangle.

30-60-90 Triangle

The sides of a 30-60-90 triangle are in a ratio of x : x V 3 : 2x.

45-45-90 Triangle
The sides of a 45-45-90 triangle are in a ratio of x : x : x V 2

Polygon:
A polygon is a plane figure that is bounded by a closed path made of straight line segments. These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices or corners.

convex polygon is a simple polygon whose interior angles are convex.
* Every internal angle is less than 180 degrees.
* Every line segment between two vertices remains inside or on the boundary of the polygon.

Concave polygons:
A concave polygon will always have an interior angle with a measure that is greater than 180 degrees.

A quadrilateral is a polygon with four sides or edges and four vertices or corners:

A parallelogram is a quadrilateral with two sets of parallel sides.
Equivalent conditions are that opposite sides are of equal length.
Opposite angles are equal.
The diagonals bisect each other.
Parallelograms also include the square, rectangle, rhombus and rhomboid.

Kite: two adjacent sides are of equal length and the other two sides also of equal length.
Trapezium : two opposite sides are parallel.
Cyclic quadrilateral: the four vertices lie on a circumscribed circle.

Rectangle:

Rectangle normally refers to a quadrilateral with four right angles.

Properties:
• All angles are 90 degrees.
• Opposite sides are equal in length.
• Opposite sides are parallel.
• Diagonals are equal in length and bisect each other.
If a rectangle has length l and width w
• it has area A = lw
• perimeter P = 2l + 2w = 2(l + w)
• and each diagonal has length v(l^2+w^2).

Rhombus:
A rhombus or rhomb is a quadrilateral whose four sides all have the same length.
A rhombus is an equilateral quadrilateral.
Every rhombus is a parallelogram, and a rhombus with right angles is a square.

Properties
Every rhombus has two diagonals connecting opposite pairs of vertices and two pairs of parallel sides. Any rhombus has the following two properties:

1. Opposite angles of a rhombus have equal measure.
2. The two diagonals of a rhombus are perpendicular.

The first property implies that every rhombus is a parallelogram. A rhombus therefore has all of the properties of a parallelogram:
opposite sides are parallel.
the two diagonals bisect one another.

Any quadrilateral whose two diagonals are perpendicular is called a kite. Every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus.

Square:
a square is a regular quadrilateral. This means that it has four equal sides and four equal angles (90 degree angles, or right angles).

A square is both a rhombus (equal sides) and a rectangle (equal angles) and therefore has all the properties of both these shapes, namely:

The perimeter of a square whose sides have length s is
P=4s
and the area is
A=s^2

Properties:
* The diagonals of a square bisect each other.
* The diagonals of a square bisect its angles.
* The diagonals of a square are perpendicular.
* Opposite sides of a square are both parallel and equal.
* All four angles of a square are equal. (Each is 360/4 = 90 degrees, so every angle of a square is a right angle.)
* The diagonals of a square are equal.
* If the diagonals of a rhombus are equal, then that rhombus must be a square.
* A square can also be defined as a rectangle with all sides equal, or a rhombus with all angles equal, or a parallelogram with equal diagonals that bisect the angles.
* If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths), then it is a square.
* If a circle is circumscribed around a square, the area of the circle is π / 2 (about 1.57) times the area of the square.
* If a circle is inscribed in the square, the area of the circle is π / 4 (about 0.79) times the area of the square.
* A square has a larger area than any other quadrilateral with the same perimeter.