quadrilaterals

MATHS PROJECTMATHS PROJECTQuadrilateralsQuadrilateralsMATHS PROJECTMATHS PROJECTQuadrilateralsQuadrilaterals

- Shaunak Bhimani- Shaunak Bhimani

IX-BIX-B

Definition

• A plane figure bounded by four line segments AB,BC,CD and DA is called a quadrilateral.

A B

D C*Quadrilateral

I have exactly four sides.

In geometry, a quadrilateral is a polygon with four sides and four vertices. Sometimes, the term

quadrangle is used, for etymological symmetry with triangle, and sometimes tetragon for consistence

with pentagon.

There are over 9,000,000 quadrilaterals. Quadrilaterals are either simple (not self-

intersecting) or complex (self-intersecting). Simple quadrilaterals are either convex or concave.

Taxonomic ClassificationThe taxonomic classification of quadrilaterals is illustrated by the following graph.

Types of Quadrilaterals

• Parallelogram

• Trapezium

• Kite

I have:2 sets of parallel sides2 sets of equal sidesopposite angles equaladjacent angles supplementarydiagonals bisect each otherdiagonals form 2 congruent triangles

Parallelogram

Types of Parallelograms

*RectangleI have all of the properties of the

parallelogram PLUS- 4 right angles

- diagonals congruent

*RhombusI have all of the properties of the

parallelogram PLUS- 4 congruent sides- diagonals bisect

angles- diagonals

perpendicular

*SquareHey, look at me!I have all of the properties of the

parallelogram AND the rectangle AND

the rhombus.I have it all!

Is a square a rectangle?

Some people define categories exclusively, so that a rectangle is a quadrilateral with four right angles that is not a square. This is appropriate for everyday use of the words, as people

typically use the less specific word only when the more specific word will not do. Generally a rectangle which isn't a square is

an oblong.But in mathematics, it is important to define categories

inclusively, so that a square is a rectangle. Inclusive categories make statements of theorems shorter, by eliminating the need for tedious listing of cases. For example, the visual proof that

vector addition is commutative is known as the "parallelogram diagram". If categories were exclusive it would have to be known as the "parallelogram (or rectangle or rhombus or

square) diagram"!

Trapezium

I have only one set of parallel sides. [The median of a trapezium is parallel to the bases and equal to one-half the sum of the

bases.]

Trapezoid Regular Trapezoid

It has two pairs of sides.Each pair is made up of adjacent sides (the sides meet) that are equal in length. The angles are equal where the pairs meet. Diagonals (dashed lines) meet at a right angle, and one of the diagonal bisects (cuts equally in half) the other.

Kite

Cyclic quadrilateral: the four vertices lie on a circumscribed circle. Tangential quadrilateral: the four edges are tangential to an inscribed circle. Another term for a tangential polygon is inscriptible. Bicentric quadrilateral: both cyclic and tangential.                                                                                                    

Some other types of quadrilaterals

Angle Sum Property Of Angle Sum Property Of QuadrilateralQuadrilateral

The sum of all four angles of a quadrilateral is

360.. A

B C

D

1

23 4

6

5

Given: ABCD is a quadrilateral

To Prove: Angle (A+B+C+D) =360.

Construction: Join diagonal BD

Proof: In ABD

Angle (1+2+6)=180 - (1)

(angle sum property of )

In BCD

Similarly angle (3+4+5)=180 – (2)

Adding (1) and (2)

Angle(1+2+6+3+4+5)=180+180=360

Thus, Angle (A+B+C+D)= 360

The Mid-Point TheoremThe line segment joining the mid-points of two sides of a triangle is parallel to the third side and is half of it.

Given: In ABC. D and E are the mid-points of AB and AC respectively and DE is joined

To prove: DE is parallel to BC and DE=1/2 BC

1

3

2

4

A

D E F

CB

Construction: Extend DE to F such that De=EF and join CFProof: In AED and CEFAngle 1 = Angle 2 (vertically opp angles)AE = EC (given)DE = EF (by construction)Thus, By SAS congruence condition AED= CEFAD=CF (C.P.C.T)And Angle 3 = Angle 4 (C.P.C.T)But they are alternate Interior angles for lines AB and CFThus, AB parallel to CF or DB parallel to FC-(1)AD=CF (proved)Also AD=DB (given)Thus, DB=FC

Thus, the other pair DF is parallel to BC and DF=BC (By construction E is the mid-pt of DF)

Thus, DE=1/2 BC

THE END- Shaunak Bhimani

IX-BROLL NO. 04