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A quadrilateral

Définition:

 

A quadrilateral is a polygon with four sides. Quadrilaterals can be simple and complex, and those simple can be convex and concave.

 

The inscriptible quadrilateral is a quadrilateral, the vertexes of which belong to a circle; the square, rectangle and isosceles trapezium are inscriptible quadrilaterals;

The tangential quadrilateral is the quadrilateral that can be inscribed in a circle. Pithot theorem refers to this type of quadrilaterals: "A convex quadrilateral is tangential if and only in case the sum of opposite sides’ lengths are equal".

The convex quadrilaterals:

The convex quadrilaterals are the most known quadrilaterals. By definition, a quadrilateral is convex when the supporting line of each side has the following property: in one of the opened half-planes, determined by it, there are two vertexes of the quadrilateral. Another definition, less rigorous but more intuitive, is that extension of any side does not make an intersection with any other side.

 

A trapezius two opposite sides are parallel.

 

Trapezium is a particular case of convex quadrilateral, having two opposite sides parallel and the other two are non-parallel. Parallel sides of a trapezium are called bases. The distance between those two bases is called trapezium’s height;

• the scalene trapezium has those two non-parallel sides of different lengths and none of them form a right angle with the base;

• the right trapezium one of the non-parallel sides is perpendicular on those two bases;

• the isosceles trapezium two sides are parallel and the other two are congruent; presents the property that the diagonals are congruent;

   Properties:

the angles adjacent to base are congruent;

the opposite angles are compplementary;

the diagonals are congruent;

in case the diagonals are perpendicular, the height is equal to middle line, and the area is equal to the square of heighti;

the area is equal to the product of middle line and height.

   Reciprocal theoremes:

In a trapezium, if the angles adjacent to base are congruent, then the trapezium is isosceles;

In a trapezium, if the diagonals are congruent, then the trapezium is isosceles;

 

The parallelogram opposite sides are parallel and congruent two by two.

 

   Proprietes :

The opposite sides are congruent;

Two adjacent angles are supplementary (their sum is equal to 180°);

Its diagonals are cut in congruent segments (in halfs);

In a parallelogram, the opposite angles are congruent, and the adjacent angles are supplementary;

Area of a parallelogram is equal to doubled area of the triangle formed by two adjacent sides and their opposed diagonal;

Area of a parallelogram is equal to the product of the legths of two adjacent sides and the sine of the one of parallelogramțs angles;

   Reciprocal theoremes:

In a convex quadrilateral, if the opposed sides are congruent two by two, then the quadrilateral is a parallelogram;

In a convex quadrilateral, if two opposed sides are parallel and congruent, then the quadrilateral is a parallelogram;

In a convex quadrilateral, if the opposed angles are congruent, then the quadrilateral is a parallelogram;

In a convex quadrilateral, if diagonals have the same midpoint, then the quadrilateral is a parallelogram;

 

A rhombus is a parallelogram with all sides equal; its diagonals are perpendicular with each other

 

All parallelogram’s proprieties are vaild for rhombus, because it is a particular parallelogram.

A particular case of rhombus is square, where all angles are equal (congruent). A rhombus with an angle of 90 degrees is a square.

   Properties:

the opposite sides are parallel and congruent;

the opposite angles are congruent (equal), and the adjacent angles are suplementary;

the diagonals are cut in congruent segments (halfs);

the area is equal to the doubled area of a triangle formed by two adjacent sides and their opposite diagonal;

the area is equal to the product of diagonals divided by two;

   Additionally:

all sides are congruent;

diagonals are (reciprocally) perpendicular;

diagonals are also the bisectors of the angles;

the perimeter is the length multiplied by 4;

the area is equal to half of the product of diagonals;

the area is equal to the product of the square of a side and sine of one of the rhombus’ angles;

the area is four times larger than the area of a triangle formed by a side and two semi-diagonals;

 

A rectangle is a quadrilateral (polygon with four sides), a particular type of parallelogram where all angles are right.

 

   Proprietes:

opposite sides are parallel and congruent;

the diagonalas are congruent;

the angles are congruent and they measure 90 deggrees;

the larger side is called length (L), and the shorter one is called width (W);

the area is equal to the product of the length and width;

the area is four times greater than the area of the triangle formed by a side and two diagonals;

the perimeter of a rectangle is equal to the doubled sum of the length and width;

the square represents a particular case of a rectangle, where the length and width are equal.

 

A square is a rhombus with all angles being right or a rectangle with all sides being equal; presents the property that the diagonals are congruent and perpendicular.

 

It is a particular case of rectangle (a rectangle with all adjacent sides being equal) and of rhombus (a rhombus with all angles being right).

   Proprietes:

the opposite sides are parallel;

all sides are equal;

all angles are right;

adjacent sides are perpendicular;

area is equal to the square of a side;

area is equal to the product of diagonals divided by 2;

perimeter is equal to the side multiplied by 4;

diagonals are congruent and perpendicular;

diagonals are the bisectors of the angles;

midpoints of the sides form an another square;

has 4 axis of symetry;

 

Informations contextuelles: unité de mesure contemporaine et archaïque de la superficie