Quadrilateral Properties | Trapezium, parallelogram, Rhombus
What is Quadrilaterals
A Quadrilaterals is defined as a simple closed figure bounded by four lines in plane. If any four points are not colliner, then we obtain a closed figure.
There above all types of figures are know as quadrilaterals. We can easily draw many more quadrilaterals and we can identify many around us.
Terminology related to Quadrilaterals :
Sides and adjacent sides :
The four line segments AB, BC, CD & DA are known as sides of quadrilateral. Two sides of quadrilaterals which have a common end point are called the adjacent sides.
Thus ( AB, BC) ; ( BC , CD ) ; ( CD , DA ) ; ( DA , AB ) are four pairs of adjacent sides of the quadrilateral of ABCD.
Vertices of the quadrilateral :
The points A, B, C & D are the Vertices of the quadrilateral ABCD.
Angles and adjacent sides :
The four angles ∠DAB , ∠ABC, ∠BCD & ∠CDA are known as angles of quadrilateral. These angles can be also denoted as ∠A , ∠B, ∠C & ∠D respectively.
Two angles of the a quadrilateral having a common side are called its adjacent angles. Thus (∠A , ∠B ) ; ( ∠B, ∠C ) ; (∠C , ∠D ) ; ( ∠D, ∠A) are four pairs of adjacent angles of the quadrilateral of ABCD.
Two angles of a quadrilateral which are not adjacent angles are the opposite angles . Thus (∠A , ∠C ) ; ( ∠B, ∠D ) are two pairs of opposite angles of the quadrilateral of ABCD.
Diagonals of the quadrilateral:
The line segment joining the opposite vertices of a quadrilateral is called a Diagonals of the quadrilateral.
Here AC and BD are two diagonals of the quadrilateral ABCD.
Perimeter of the quadrilateral:
Quadrilateral perimeter mean, The sum of length of all the four sides of quadrilateral.
Here Perimeter of the quadrilateral ABCD = AB + BC + CD + DA.
Types of quadrilaterals with formulas :
Properties and formulas of Trapezium (Trapezoid) :
The quadrilateral having one pair of opposite sides parallel to each other is called a trapezium.
Here AD || BC , Height from base AD to base BC is ” h” and length of AD = a and BC = b
Area of the trapezoid = 1/2 x sum of parallel sides x height
Area of Trapezium (Trapezoid) ABCD = (1/2 ) (a + b) h.
- If the non -parallel sides are equal then diagonals will be too.
- Median of trapezium = 1/2 x sum of parallel sides ( median is the line equidistant from the parallel sides). Here EF is median of trapezium ABCD . EF = (a + b) /2. (here AE = EB & DF= FC).
Isosceles trapezium :
The quadrilateral having one pair of opposite sides parallel to each other and other pair of sides are equal then it is called an Isosceles trapezium.
Parallelogram Properties and formulas:
If both pairs of opposite sides of the quadrilateral are parallel, such a quadrilateral is called parallelogram.
Area of parallelogram = Base x height = bh
Perimeter of parallelogram = 2 ( b+ c )
Area of parallelogram = product of any two adjacent sides x sine of the included angle = b x c x sin B.
- The opposite sides of a parallelogram are equal in length. Thus AB = DC and AD = BC.
- Opposite angles of a parallelogram are congruent (equal measure).
- The diagonals of a parallelogram bisect each other. Here BE = ED and AE = EC.
- Each diagonal of a parallelogram divides it into two triangles of the same area. Here area of ΔABC = Area of ΔACD and area of ΔABD = area of ΔBCD.
- Bisectors of the angles of parallelogram form a rectangle.
- A parallelogram inscribed in circle is a rectangle.
- A parallelogram circumradius about a circle is a rhombus.
- The sum of the squares of the diagonals is equal to the sum of the square of the four sides.
- Here AC2 + BD2 = AB2 + BC2 + CD2 + DA2 = 2 b2 + 2 c2
Properties and formulas of Rectangle:
A parallelogram having any one of its angle as right angle ( 90°) is a rectangle.
Area of the rectangle = bh
Perimeter of the rectangle = 2 (b + h)
Length of diagonal ( l ) = √ b2 + h2
- The diagonals of a rectangle are equal and bisect each other.
- The opposite sides of a rectangle are parallel and opposite sides of a rectangle are congruent. Here AD || BC and AB || DC.
- All four angles of a rectangle are right angles. Here ∠A =∠B = ∠C = ∠D = 90°.
- All rectangles are parallelograms but the reverse is not possible.
Properties and formulas of Rhombus:
Area of the Rhombus = 1/2 x product of diagonals x sine of the angle between them.
Area of the Rhombus ABCD = (1/2) d1 d2 ( Diagonals are right angle so sin 90° = 1)
Here Area of the Rhombus ABCD = bh
Perimeter of rhombus = 4b
- All the sides of a rhombus are congruent. Here AB = BC = CD = DA = b.
- Opposite sides of a rhombus are parallel. Here AB || DC & AD || BC.
- The diagonals of a rhombus bisect each other at right angles. Here ∠AED = ∠AEB = ∠BEC = ∠DEC = 90°.
- Opposite internal angles of a rhombus are congruent (equal in size). Here ∠ABC = ∠ADC and ∠BAD = ∠DCB.
- The sum of any two consecutive internal angles of a rhombus equal to 180°. Here ∠A + ∠B = ∠B + ∠C = ∠C +∠D = ∠D + ∠A = 180°.
- All rhombuses are parallelograms but the reverse is not possible.
- A rhombus may or may not be a square but all square but all squares are rhombuses.
Properties and formulas of Square :
A parallelogram having equal adjacent sides and all angles of right angle (90°) is called square.
Here length of the side for square ABCD = a
Length of diagonal = d = √2 a
Area of the square = b2
Area of the square ABCD = (1/2) d2
Perimeter of the square = 4b
- All four sides of a square are congruent.
- Opposite sides of a square are parallel. Here AB || DC & AD || BC.
- The diagonals of a square are equal. Here AE = BD
- The diagonals of a square bisect each other at right angles. Here ∠AED = ∠AEB = ∠BEC = ∠DEC = 90°.
- All angles of a square are 90° . Here ∠A = ∠B = ∠C = ∠D = 90°.
- All squares are belongs to a special kind of rectangles where all the sides have equal length.
- Side is the diameter of the inscribed circle.
- Diagonal is the diameter of the circumradius circle. Here Diameter = √2 a & circumradius = a / √2
Area of Kite:
The quadrilateral having two pairs of equal adjacent sides is called kite.
Here BC = DC = a & AB = AD = b
d1 is the length of a diagonal.
d2 is the length of the other diagonal.
Area of kite = (1/2) d1 d2.
Some Important points in quadrilaterals
- Parallelograms, trapeziums, rhombuses, rectangles and squares are all quadrilaterals.
- Parallelograms, rhombuses, rectangles and squares are all trapeziums.
- Rhombuses, rectangles and squares are all Parallelograms.
- All squares are Rhombuses but the converse is not true.
- all squares , rhombuses are kites.
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