Maths porject work - quadrilaterals - nihal gour

MATHS PORJECT WORK

QUADRILATERALS

MADE BY : -

Nihal Gour

SUBMITTED TO :-

Mr. D. K. Chourasia

IX B

The word quadrilateral is made of the words quad (meaning "four") and lateral (meaning "of sides").

This Gives us a simple definition about a quadrilateral : A polygon (a closed figure made of only line segments) with four side is called a quadrilateral.

INTRODUCTION

4 vertices

A.

B.

C.

D.

• 4 sides

• 4 angles

QUADRILATERALS

QUADRILATERALS

A.

B.

C.

D.

360o

The sum of ALLALL the angles of a

quadrilateral is 360o

The sum of ALLALL the angles of a

quadrilateral is 360oC

.

A.

D.

B.

QUADRILATERALS

A.

B.

C.

D.

360o

The sum of ALLALL the angles of a

quadrilateral is 360o

The sum of ALLALL the angles of a

quadrilateral is 360o

Angle Sum Property Of Quadrilateral (In Detail)

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

DifferEnttypes

ofQUADRILATERALS

The TRAPEZIUMThe TRAPEZIUM

• One pair of opposite sides are parallel

The TRAPEZIUMThe TRAPEZIUM

The PARALLELOGRAM

The PARALLELOGRAMThe PARALLELOGRAM

• Opposite sides are equal

• Opposite sides are parallel• Opposite angles are equal

• Diagonals bisect each other

The RHOMBUS

The RHOMBUS

All sides are equal• Opposite sides are parallel• Opposite angles are equal

• Diagonals bisect each other at 90°

The RECTANGLE

The RECTANGLE

Opposite sides are equal• Opposite sides are parallel• All angles are right angles (90o)• Diagonals are equal and bisect one another

The SQUARE

The SQUARE

All sides are equal• Opposite sides are parallel

• All angles are right angles (90o)

• Diagonals are equal and bisect one another at right angles

•Each pair adjacent sides (the sides meet) are equal in length.

•The angles are equal where the pairs meet.

•Diagonals (dashed lines) meet at a right angle

•The longer diagonal bisects (cuts equally in half) the shorter diagonal.

Kite

A

B

C

D AC bisected BD

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

Quadrilaterals Flow Chart (Simpler)

General Quadrilateral

4 sides, 4 anglesTrapezoid

Only 1 pair of parallel sides

Parallelogram

Opposite sides are parallel and congruent

Rectangle

A parallelogram with 4 right angles

Rhombus

A parallelogram with 4 congruent sides

Square

A rectangle with 4 congruent sides

Kite

Note that…….

A square, rectangle and rhombus are all parallelograms.

A square is a rectangle and also a rhombus. A parallelogram is a trapezium. A kite is not a parallelogram. A trapezium is not a parallelogram (as only

one pair of opposite sides is parallel in a trapezium and we require both pairs to be parallel in a parallelogram).

A rectangle or a rhombus is not a square.

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

Some Properties of Parallelogram,

Rectangle, Rhombus and

Square

• A diagonal of a parallelogram divides it into two congruent triangles.

• In a parallelogram,(i) opposite sides are equal (ii) opposite angles are equal(iii) diagonals bisect each other

• A quadrilateral is a parallelogram, if(i) opposite sides are equal, or (ii) opposite angles are equal, or(iii) diagonals bisect each other, or(iv) a pair of opposite sides is equal and parallel

• Diagonals of a rectangle bisect each other and are equal and vice-versa.

• Diagonals of a rhombus bisect each other at right angles and vice-versa.

• Diagonals of a square bisect each other at right angles and are equal, and vice-versa.

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 ABCD 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 CF

Proof: 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 construction E is the mid-pt of DF)

Thus, DE=1/2 BC