MATHS PORJECT WORK QUADRILATERALS
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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
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