479f3df10a8c0Mathsproject-quadrilaterals (1)

7/31/2019 479f3df10a8c0Mathsproject-quadrilaterals (1)

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MATHS PROJECT MATHS PROJECT QuadrilateralsQuadrilateralsMATHS PROJECT MATHS PROJECT QuadrilateralsQuadrilaterals

ShivaniShivani

IX-AIX-A


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Definition

A plane figure bounded by four linesegments AB,BC,CD and DA is called aquadrilateral.

A B

D C

*QuadrilateralI have exactly four sides.


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

is used, for etymological symmetry with triangle, andsometimes 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.
http://en.wikipedia.org/wiki/Convex_polygonhttp://en.wikipedia.org/wiki/Concave_polygonhttp://en.wikipedia.org/wiki/Concave_polygonhttp://en.wikipedia.org/wiki/Convex_polygon


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Taxonomic ClassificationThe taxonomic classification of quadrilaterals is illustrated by thefollowing graph.
http://en.wikipedia.org/wiki/Image:Quadrilateral_hierarchy.pnghttp://en.wikipedia.org/wiki/Image:Quadrilateral_hierarchy.pnghttp://en.wikipedia.org/wiki/Image:Quadrilateral_hierarchy.png


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Types of Quadrilaterals Parallelogram

Trapezium

Kite


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http://en.wikipedia.org/wiki/Image:Quadrilateral.png


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I have:2 setsof parallel sides

2 sets of equal sides

opposite angles equaladjacent angles supplementarydiagonals bisect each otherdiagonals form 2 congruent triangles

Parallelogram


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Types of Parallelograms

*RectangleI have all of theproperties of the

parallelogram PLUS- 4 right angles

- diagonals congruent

*RhombusI have all of the

properties of theparallelogram PLUS- 4 congruent sides

- diagonals bisect angles- diagonals perpendicular


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

properties of theparallelogram AND the

rectangle AND therhombus.

I have it all!


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Is 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 use theless 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 , sothat a square is a rectangle. Inclusive categories make statements oftheorems shorter, by eliminating the need for tedious listing of cases.

For example, the visual proof that vector addition is commutative isknown as the "parallelogram diagram ". If categories were exclusive it

would have to be known as the "parallelogram (or rectangle orrhombus or square) diagram"!


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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


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It 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. Diagonals (dashed lines) meetat a right angle, and one of the diagonal bisects(cuts equally in half) the other .

Kite
http://en.wikipedia.org/wiki/Image:Quadrilateral.png


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Cyclic quadrilateral : the four verticeslie on a circumscribed circle.

Tangential quadrilateral : the fouredges are tangential to an inscribedcircle. Another term for a tangential

polygon is inscriptible .Bicentric quadrilateral : both cyclicand tangential.

Some other types ofquadrilaterals
http://en.wikipedia.org/wiki/Cyclic_quadrilateralhttp://en.wikipedia.org/w/index.php?title=Tangential_quadrilateral&action=edithttp://en.wikipedia.org/w/index.php?title=Bicentric_quadrilateral&action=edithttp://en.wikipedia.org/wiki/Image:Quadrilateral.pnghttp://en.wikipedia.org/w/index.php?title=Bicentric_quadrilateral&action=edithttp://en.wikipedia.org/w/index.php?title=Tangential_quadrilateral&action=edithttp://en.wikipedia.org/wiki/Cyclic_quadrilateral


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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 quadrilateral

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

Construction: Join diagonal BD


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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


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The Mid-Point TheoremThe line segment joining the mid-points of two sides ofa 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 respectivelyand DE is joined

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

1

3

2

4

A

D E F

CB


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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 -(2)From (1) and(2)DBCF is a gm

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

Thus, DE=1/2 BC


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THE END shivaniIX-A

ROLL NO. 29Have a nice day mam..

479f3df10a8c0Mathsproject-quadrilaterals (1)

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