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MATHS PROJECT Quadrilaterals. - Monica Sant IX-A. Definition. A plane figure bounded by four line segments AB,BC,CD and DA is called a quadrilateral. A. B. C. D. In geometry, a quadrilateral is a polygon with four

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

A

B

C

D

slide3

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.

slide4

Taxonomic Classification

The taxonomic classification of quadrilaterals is illustrated by the

following graph.

types of quadrilaterals
Types of Quadrilaterals
  • Parallelogram
  • Trapezium
  • Kite
slide10

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"!

slide11

Trapezium

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

slide12

Kite

Ithas 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.

slide13

Some other types of quadrilaterals

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.

angle sum property of quadrilateral

Angle Sum Property Of Quadrilateral

The sum of all four angles of a quadrilateral is 360..

A

D

1

6

5

2

4

3

B

C

Given: ABCD is a quadrilateral

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

Construction: Join diagonal BD

slide15

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

slide16

The Mid-Point Theorem

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

A

3

1

E

D

F

2

4

B

C

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

slide17
Construction: Extend DE to F such that De=EF and join CF

Proof: In AED and CEF

Angle 1 = Angle 2 (vertically opp angles)

AE = EC (given)

DE = EF (by construction)

Thus, By SAS congruence condition AED= CEF

AD=CF (C.P.C.T)

And Angle 3 = Angle 4 (C.P.C.T)

But they are alternate Interior angles for lines AB and CF

Thus, 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 construction E is the mid-pt of DF)

Thus, DE=1/2 BC