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Quadrilaterals

T here are hardly any questions based on quadrilaterals alone. They usually find an appearance
along with some theory of triangles or circles, as we have seen in the questions based on
triangles. So the following test briefly captures the salient points of the various quadrilaterals…

Parallelogram
Any one of the following conditions is sufficient in itself to define a parallelogram
Both the pairs of opposite sides are parallel
One pair of opposite sides are parallel and equal
The diagonals of the quadrilateral bisect each other
Thus, if in any quadrilateral the diagonals bisect each other, it necessarily has to be
a parallelogram.

Further it would be worthwhile to keep the following in mind about a parallelogram…

Opposite sides are equal in length
Opposite angles are equal and adjacent angles are supplementary.
Area of a parallelogram = base × height
The areas of triangles ABC and ADB are equal as they stand on the same base and
have the same height.
Since the diagonals bisect each other, BO will be a median in triangle ABC and
similar results exists for other half of diagonals. Thus one can apply Apollonius
Theorem in questions involving lengths of the diagonals.
A common misconception is that the diagonals are also the angle bisectors. The
diagonals need not be the angle bisectors in a parallelogram.

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Rhombus
In a parallelogram, when the lengths of the adjacent sides are equal, it becomes
a rhombus. Thus rhombus is a specific case of parallelogram and as such all the
properties of a parallelogram would also be valid for a rhombus.

Additionally because of the specific case of adjacent sides being equal, we have the
following additional properties..
The diagonals are going to bisect each other at right angles.
The diagonals are going to be the angle bisectors as well.
1
The area of a rhombus is d1d2 where d1 and d2 are the lengths of the
2
diagonals. (In fact this formula for area is valid for any quadrilaterals where
the diagonals intersect at right angles e.g. kite). The earlier formula i.e.
base × height, is still valid as rhombus is also a parallelogram.

Rectangle
In a parallelogram when the adjacent sides are perpendicular to each other, it
becomes a rectangle. Thus a rectangle is also a specific case of parallelogram and as
such all the properties of a parallelogram would also be valid for a rectangle as well.
The only additional property as distinct from a parallelogram is that the diagonals
would become equal in length. The diagonals need not be the angle bisector (this will
happen only when adjacent sides are equal)

Square
In this case, the adjacent sides are perpendicular to each other and are equal to
each other as well. Thus a square is a rhombus, is also a rectangle and obviously
parallelogram being the parent figure, a square is also a parallelogram.
Thus, a square will have all the properties of a rhombus as well, specifically,
diagonals bisecting at right angles and diagonals being the angle bisector. Similarly,
1
while the area of a square is (side)2, it is also × (diagonal)2 , using the formula for
2
area of a rhombus.

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Trapezium
A distinct figure as compared to the above is a trapezium where only one pair of
opposite sides is parallel. The other pair of opposite sides, which are not parallel, are
called oblique sides.

In a trapezium…
1. The opposite angles are not equal. Just the allied interior angles between
the parallel lines have to be supplementary.
2. Line joining the mid-points of the oblique sides is parallel to the parallel
sides and its length is the arithmetic mean of the lengths of the parallel sides.
1
3. Area = × (sum of parallel sides) × height .
2
A specific type of trapezium is an Isosceles Trapezium. In this the lengths of the
oblique sides are equal. Because of this, in an isosceles trapezium…
The base angles are equal and so are the other two angles
The diagonals become equal in length

Cyclic Quadrilateral
A cyclic quadrilateral is one in which the four vertices of the quadrilateral lie on a
circle. In a cyclic quadrilateral the opposite angles are supplementary (the reason
for this will be learnt in the chapter on circles). Whenever it is mentioned that a
quadrilateral is cyclic, most often the above property will be used. Another feature
worth remembering for a cyclic quadrilateral is that the exterior angle is equal to the
sum of remote interior angle.

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Polygons
Any closed figure whose sides are straight lines is a polygon. Examples…

If all the vertices lie on the same side of the line containing each of the sides of
the polygon, then it is a Convex Polygon. If the above is not the case, it will be a
Concave Polygon.
As far as management entrance exams go, we will focus mainly on convex polygon.
So unless specified otherwise, assume the polygon to be convex.
Further polygons are named depending on the number of sides, as follows:
3 sides: Triangle 4 sides: Quadrilateral 5 sides: Pentagon
6 sides: Hexagon 7 sides: Heptagon 8 sides: Octagon
9 sides: Nonagon 10 sides: Decagon 12 sides: Dodecagon

Regular Polygon:
A regular polygon is one in which all the sides are of equal length. Consequently all
the interior angles of a regular convex polygon would also be equal.

Interior and Exterior Angle:

The following figure depicts a pair of interior and exterior angle for a triangle and a
hexagon.
Exterior angle is formed by extending a side of the polygon and the angle formed by
the extended side with the adjacent side is called the exterior angle.

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A polygon of n sides has n exterior angles. Thus, a triangle will have three exterior
angles. The following figure depicts all the three exterior angles. Please note that
either of the three triplets can be taken as a set of exterior angles, and it is wrong to
state that there are 6 exterior angles.

Sum of all exterior angles of any convex polygon is 360o . This is valid for all convex
polygons, irrespective of the number of sides.
The above can be used to find the sum of all interior angles of any convex polygon.
We know a pair of interior and exterior angles would add up to 180o . In a polygon
of n sides, there would be n pairs of interior and exterior angle and their sum would
be n × 180o . Excluding the sum of all exterior angles i.e. 360, the sum of all interior
angles will be n × 180o − 360o = (n − 2) × 180o or as it is usually expressed ( 2n − 4 ) × 90o

Thus, one should remember the following facts about Regular polygons…
Polygon Each exterior angle Each interior angle Sum of interior angles
360 o o
Triangle = 120o 60 180
3

360 o o
Quadrilateral = 90o 90 360
4

360 o o
Pentagon = 72o 108 540
5

360 o o
Hexagon = 60o 120 720
6

360 o o
Octagon = 45o 135 1080
8

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Of these polygons, the next popular one after triangles and quadrilaterals, are
hexagons. So please keep the following in mind while dealing with a regular
hexagon…
A regular hexagon can be considered as 6 equilateral triangles placed side by side….

3 2
Thus, if a is the side of the hexagon, then the area of the hexagon = 6 × a
4

Exercise 8
1. The lengths of the diagonals of a parallelogram are 16 units and 30 units. If the length of one
side of the parallelogram is 17 units, what is the perimeter of the parallelogram?
(1) 64 (2) 66 (3) 68 (4) 72
2. In a rhombus, if the two diagonals measure 24 units and 32 units, find the perimeter of the
rhombus.
(1) 40 (2) 80 (3) 120 (4) 160
3. In rectangle ABCD, points P, Q, R and S divide the sides AB, CB, CD and AD in the ratio 2 : 3.
Find the ratio of the area of quadrilateral PQRS to the area of rectangle ABCD.
(1) 16 : 25 (2) 19 : 25 (3) 4 : 9 (4) 12 : 25
4. In parallelogram ABCD, the midpoints of AB, BC, CD and AD are joined to form another
quadrilateral PQRS. If the area of quadrilateral PQRS is a sq units, what is the area of the
parallelogram (in terms of a)?
(1) a (2) 2a (3) 4a (4) 8a
5. Square ABCD, with side = 3 cm, is rotated by 45 degree keeping its center fixed to result into
another square PQRS. What is the area of the region common to the two squares?

1+ 2 1+ 2 2 2+2 2
(1) 7 (2) 9 × (3) 9 × (4) 9 ×
2+ 2 3+2 2 3+2 2

6. In parallelogram ABCD, the bisector of angle ABC intersects AD at point P. If l(PD) = 5, l(BP) =
6, and l(CP) = 6, find the length of AB.
(1) 3 (2) 4 (3) 5 (4) 6
7. In isosceles trapezium ABCD, AB and CD are the parallel sides and have lengths equal to 16 cm
and 10 cm. If the length of oblique sides is 5 cm, find the area of the trapezium.
(1) 65 (2) 48 (3) 52 (4) 56

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8. In a trapezium ABCD with AB || CD, EF is a line parallel to the parallel sides with E and F
lying on AD and BC such that the perimeter of trapezium ABFE and EFCD are equal. If
AB = 18, BC = 6, CD = 13 and AD = 4, find the ratio AE : ED.
(1) 1 : 2 (2) 1 : 3 (3) 1 : 4 (4) 2 : 3
9. If the ratio of interior angles of two regular polygons is 75 : 78 and the difference in the number
of sides of the two polygons is 3, then find the ratio of the number of sides of the polygons.
(1) 5 : 4 (2) 4 : 3 (3) 4 : 5 (4) 3 : 4
10. If the difference between the sum of all interior angles of two polygons is 720, find the difference
between the number of sides of the two polygons.
(1) 4 (2) 5 (3) 6 (4) 7
11. Each side of a given polygon is parallel to either the X-axis or the Y-axis. A corner of such a
polygon is said to be convex if the internal angle is 90 degrees or concave if the internal angle is
270 degrees. If the number of convex corners is 25, find the number of concave corners.
(1) 21 (2) 25 (3) 29 (4) Cannot be determined
12. In a hexagon of unit sides, three alternate vertices are joined to form an equilateral triangle
within the hexagon. Find the area of the triangle so formed?

3 3 3 3 3
(1) 3 (2) (3) (4)
2 4 4

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