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Quadrilaterals and their Properties

A quadrilateral is a figure bounded by four sides. In the figure given below ABCD is a quadrilateral. Line AC is the diagonal of the quadrilateral (denoted by d) and BE and DF are the heights of the triangles ABC and ADC respectively (denoted by h1 and h2).
 
Description: 23206.png
AC = d, BE = h, and DE = h2
  1. Area Description: 6441.png one diagonal × (sum of perpendiculars to the diagonal from the opposite vertexes) Description: 6433.png d (h1 + h2)
  2. Area Description: 6426.png product of diagonals × sine of the angle between them
  3. Area of the cyclic quadrilateral Description: 6419.png where abc and d are the sides of quadrilateral and s = semi-perimeter Description: 6412.png
  4. Brahmagupta’s formula For any quadrilateral with sides of length abc and d, the area A is given by Description: 6405.png
     
    Where Description: 6398.png is known as the semi-perimeter, A is the angle between sides a and d, and B is the angle between the sides b and c.

Different Types of Quadrilaterals

Parallelogram

A parallelogram is a quadrilateral when its opposite sides are equal and parallel. The diagonals of a parallelogram bisect each other.
Description: 23213.png
 
Given: AD = BC = a and AB = DC = b
BD = d
AF (height of ∆ABD) = CG (height of ∆CBD) and AE = height of the parallelogram = h ADC = θ
  1. Area = base × height
  2. Area = (any diagonal) × (perpendicular distance to the diagonal from the opposite vertex)
  3. Area = (product of adjacent sides) × (sine of the angle between them) Area = AB sin θ
  4. Area Description: 6383.png where a and b are the adjacent sides and d is the diagonal.
  5. AC2 + BD2 = 2(AB2 + BC2)
  6. The parallelogram that is inscribed in a circle is a rectangle.
  7. The parallelogram that is circumscribed about a circle is a rhombus.
  8. A parallelogram is a rectangle if is diagonals are equal.

Rectangle

Description: P-334-4.tif
 
A rectangle is a quadrilateral when its opposite sides are equal and each internal angle equals 90°. The diago­nals of a rectangle are equal and bisect each other.
Given: AD = BC = b and AB = DC = l, BD = d
  1. Area = length × breadth Area = lb
  2. Perimeter = 2 (length + breadth) Perimeter = 2 (l + b)
  3. Diagonal2 = length2 + breadth2 (Pythagoras Theorem) d2 = l2 + b2 Description: 6368.png
  4. Finding area using Brahmagupta’ Formula In this case, we know that a = c and b d, and A + B = π So, area of Rectangle
    Description: 6361.png
    Description: 11520.png
  5. The quadrilateral formed by joining the mid points of intersection of the angle bisectors of a parallelo­gram is a rectangle. 

Square

A square is a quadrilateral when all its sides are equal and each internal angle is of 90°. The diagonals of a square bisect each other at right angles (90°)
Description: P-335-1.tif
 
Given: AB = BC = CD = DA = a
BD (diagonal) = Description: 6345.png
  1. Area = (side)2 Description: 6338.pngDescription: 6331.png
    Area Description: 6324.png
  2. Using Brahmagupta’s formula to find out the area of a square:
    We know that a = b = c = d and A + B = π
    So, area of squareDescription: 6317.pngDescription: 6310.png
  3. Perimeter = 4 (side) = Perimeter = 4a

Rhombus

A rhombus is a quadrilateral when all sides are equal. The diagonals of a rhombus bisects each other at right angles (90°)
Description: P-335-2.tif
 
Given = AB = BC = CD = DA = a
AOB = BOC = COD −DOA = 90°
AC = d, (AO = OC) and BD = d2 (BO = OD) CE (height) = h
  1. Area Description: 6295.png (product of the diagonals)
    Area Description: 6288.png d1 d2
  2. Area = base × height
    Area = a × h
  3. A parallelogram is a rhombus if its diagonals are perpendicular to each other. Remember, the sum of the square of the diagonals is equal to four time the square of the side i.e., d12d22 = 4a2

Trapezium

A trapezium is a quadrilateral in which only one pair of the opposite sides is parallel
Description: P-335-3.tif

Given: AB = a and CD = b
In Fig. 1, AF (height) = h and in Fig. 2, BC (height) = h
  1. Area Description: 6271.png (sum of the parallel sides) × (distance between the parallel sides)
    Area Description: 6264.png (a + b) h
  2. The line joining the mid-points of the non-parallel sides is half the sum of the parallel sides and is known as Median.
  3. If we make non-parallel sides equal, then the di­agonals will also be equal to each other.
  4. Diagonals intersect each other proportionally in the ratio of the lengths of the parallel sides.
  5. If a trapezium is inscribed inside a circle, then it is an isosceles trapezium with oblique sides being equal.

Kite

Kite is a quadrilateral when two pairs of adjacent sides are equal and the diagonals bisect each other at right angles (90°).
Given: AB = AD = a and BC = DC = b
Description: P-335-4.tif
 
AC = d1 (AO = OC) and BD = d2 (BO = OD)
AOB = BOC = COD = DOA = 90°
  1. Area Description: 6249.png (Product of the diagonals)
    Area Description: 6242.png d1 d2

Area of Shaded Paths

Case 1 When a pathway is made outside a rectangle having length = l and breadth = b
Description: P-336-1.tif
 
ABCD is a rectangle with length = l and breadth = b, the shaded region represents a pathway of uniform width = W
Area of the shaded region/pathway = 2w (l + b − 2w)
 
Case 2 When a pathway is made inside a rectangle having length = l and breadth = b
Description: P-336-2.tif
 
ABCD is a rectangle with length = l and breadth = b, the shaded region represents a pathway of uniform width = w
Area of the shaded region/pathway = 2w (l + b + 2w)
 
Case 3 When two pathways are drawn parallel to the length and breadth of a rectangle having length = l and breadth = b
Description: P-336-3.tif
 
ABCD is a rectangle with length = l and breadth = b, the shaded region represents two pathways of a uniform
width = w
Area of the shaded region/pathway = W (l + b − w)
From the above figure we can observe that the area of the paths does not change on shifting their positions as long as they are perpendicular to each other.
 
We can conclude from here that:
  1. Every rhombus is a parallelogram but the converse is not true.
  2. Every rectangle is a parallelogram but the con­verse is not true.
  3. Every square is a parallelogram but the converse is not true.
  4. Every square is a rhombus but the converse is not true.
  5. Every square is a rectangle but the converse is not true.

Construction of new figures by joining the mid-points

Lines joining the

mid-points of

adjacent sides of

Original Figure

form

Resulting Figure

Quadrilateral

Parallelogram

Parallelogram

Parallelogram

Rectangle

Rhombus

Rhombus

Rectangle

Trapezium

Four similar ∆


Properties of diagonals

Properties Types of Quadrilaterals
Sl. No.   Square Rectangle Parallelogram Rhombus Trapezium
1 Diagonals are equal Y N N Y N
2 Diagonals bisect each other Y Y Y Y N
3 Diagonals bisect vertex angles Y N N Y N
4 Diagonals are at right angles Y N N Y N
5 Diagonals make congruent triangles Y N N Y N

Some important points

  1. In the figure given below, all the side quadrilaterals are squares and circles are inscribed in these squares. If the side of the square ABCD = a, then the side of square EFGH Description: 6181.png and the side of square MNKP Description: 6174.png In other words we can say that in order to obtain the side of the next inner square, divide the side of the immediate outer square by Description: 6167.png The same procedure will be applied for the inscribed circles i.e., we divide the radius of the immediate outer circle by Description: 6160.png to obtain the radius of the next inner circle.
     
    Description: P-337-1.tif
  2. Triangles on the same base and between the same parallel lines are equal in area.
     
    Description: P-337-2.tif
     
    Area (ABC) = Area (∆ABD) = Area (∆ABE)
     
    Description: 6137.png (AB) × h
     
    (Base AB is the same and the height ‘h’ is also the same.)
  3. If a parallelogram and a triangle are drawn on the same base and between the same parallel lines, then the area of the parallelogram is twice the area of the triangle.
     
    Area of ABCD = 2 × (Area of ∆ ABE)
     
    Description: P-337-3.tif
Example
Side AB of a rectangle ABCD is divided into four (4) equal parts as shown in the figure. Find the ratio of the area (∆XYC) and area (ABCD)?
 
Description: P-337-4.tif
Solution
Let the area of the rectangle ABCD = A
 
Area of the rectangle XYQP = A/4
 
Rectangle XYQP and ∆XYC are on the same base and between the same parallel lines.
 
 
Description: P-337-5.tif
 
Thus area (XYC) = A/8
 





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