479f3df10a8c0Mathsproject Quadrilaterals

8/3/2019 479f3df10a8c0Mathsproject Quadrilaterals

1/18

MATHS PROJECT MATHS PROJECT QuadrilateralsQuadrilateralsMATHS PROJECT MATHS PROJECT QuadrilateralsQuadrilaterals

-- Monica SantMonica Sant

IXIX--AA


2/18

D efinition

A plane figure bounded by four linesegments AB,BC,CD and D A is called a

quadrilateral.A B

D C

*QuadrilateralI

have exactly four sides.


3/18

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

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

pentagon.

There are over 9,000,000 quadrilaterals. Quadrilateralsare either simple (not self-intersecting) or complex(self-intersecting). Simple quadrilaterals are either

convex or concave.


4/18

Taxonomic ClassificationThe taxonomic classification of quadrilaterals is illustrated by thefollowing graph.


5/18

Types of QuadrilateralsParallelogram

Trapezium

Kite


6/18


7/18

I have:2 setsof parallel sides

2 sets of equal sidesopposite angles equaladjacent angles supplementarydiagonals bisect each otherdiagonals form 2 congruent triangles

Parallelogram


8/18

Types of Parallelograms

*RectangleI have all of the

properties of theparallelogram PLUS

- 4 right angles- diagonals congruent

*RhombusI have all of the

properties of theparallelogram PLUS- 4 congruent sides- diagonals bisect

angles- diagonals

perpendicular


9/18

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

properties of theparallelogram AND the

rectangle AN D therhombus.

I have it all!


10/18

I s a square a rectangle?

Some people define categories exclusively, so that a rectangle is aquadrilateral with four right angles that is not a square. This is

appropriate for everyday use of the words, as people typically usethe 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. I nclusive categories make

statements of theorems shorter, by eliminating the need for tediouslisting of cases. For example, the visual proof that vector addition iscommutative is known as the "parallelogram diagram ". I f categorieswere exclusive it would have to be known as the "parallelogram (or

rectangle or rhombus or square) diagram"!


11/18

TrapeziumI have only one set of parallel sides.

[The median of a trapezium is parallel to thebases and equal to one-half the sum of the

bases .]

Trapezoid Regular Trapezoid


12/18

I t has two pairs of sides.Each pair is made up of adjacent sides (the sidesmeet) that are equal in length. The angles are equalwhere the pairs meet. D iagonals (dashed lines) meet

at a right angle, and one of the diagonal bisects(cuts equally in half) the other .

Kite


13/18

Cyclic quadrilateral : the fourvertices lie on a circumscribed circle.

Tangential quadrilateral : the fouredges are tangential to an inscribedcircle. Another term for a tangentialpolygon is inscriptible.Bicentric quadrilateral : both cyclicand tangential.

Some other types of

quadrilaterals


14/18

Angle Sum Property OfAngle Sum Property Of

QuadrilateralQuadrilateralThe sum of all four angles of a quadrilateral is 360

..

A

B C

D

1

23 4

6

5

Given: ABCD

is a quadrilateralTo Prove: Angle (A+B+C+D ) =360 .

Construction: Join diagonal B D


15/18

Proof: I n ABD

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

(angle sum property of )I n 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


16/18

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

Given: I n ABC.D and E are the mid-points of AB and AC respectivelyand D E is joined

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

1

3

2

4

A

D E F

CB


17/18

Construction: Extend D E to F such that D e=EF and join CFProof: I n AED and CEFAngle 1 = Angle 2 (vertically opp angles)

AE = EC (given)D E = EF (by construction)Thus, By SAS congruence condition AE D = CEFAD =CF (C.P.C.T)And Angle 3 = Angle 4 (C.P.C.T)But they are alternate I nterior angles for lines AB and CF

Thus, AB parallel to CF or D B parallel to FC-(1)AD =CF (proved)Also AD =D B (given)Thus, D B=FC -(2)From (1) and(2)D BCF is a gm

Thus, the other pair D F is parallel to BC and D F=BC (By constructionE is the mid-pt of D F)

Thus, D E=1/2 BC


18/18

THE END

- MONI CA SANT IX -A

ROLL NO. 31

479f3df10a8c0Mathsproject Quadrilaterals

Documents