Trapezium

A trapezium is a quadrilateral having two parallel sides of unequal length and the other two sides are non-parallel. The parallel sides of a trapezium are called bases and the non-parallel sides of a trapezium are called legs. It is also called a trapezoid.

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In the above figure, Sides AB and CD are parallel to each other (bases) whereas AD and BC are non-parallel sides (legs). AC and BD are the two diagonals and AE is the distance between two parallel sides which is called the height of trapezium ABCD.

Shape of Trapezium

The shape of the trapezium is defined differently in different countries. Based on the sides of the trapezium, there is confusion in the distinction between the terms “trapezoid” and “trapezium”. Because of the meaning of these two words in the US and British, the definition is exactly reversed.

In the US, a trapezium is defined as a quadrilateral with no parallel sides and a trapezoid is defined as a quadrilateral with one pair of parallel sides.

In the UK, a trapezium defines a quadrilateral with one pair of parallel sides and a trapezoid is defined as a quadrilateral with no parallel sides.

Types of Trapezium

The trapezium is divided into three different categories as below:

1.    Isosceles trapezium

2.    Right trapezium

3.    Scalene trapezium

Isosceles Trapezium: 

It can be defined as a trapezium in which both legs and base angles are of the same measure.

Right Trapezium: 

A right trapezium is a trapezium having two right angles.

Scalene Trapezium: 

A scalene trapezium is a trapezium with no sides of equal measure.

Properties of Trapezium  

Every trapezium has the following properties:

1.    One pair of opposite sides are parallel.

2.    The two non-parallel sides are unequal except for isosceles trapezium.

3.    Two pairs of adjacent angles add up to 180 degrees.

4.    The diagonals intersect.

5.    The line that joins the mid-points of the non-parallel sides is always parallel to the bases or parallel sides which is equal to half the sum of the parallel sides

6. The sum of the four interior angles is 4 right angles, i.e. 3600.

7. The sum of the four exterior angles is 4 right angles, i.e. 3600.

Area of Trapezium

Area of Trapezium =\[\frac {1}{2}\] x (Sum of Parallel Sides) x (Distance Between Them).

                                 = \[\frac {1}{2}\] x (a + b) x (h)                     

Where,

“a” and “b” are the bases, and 

“h” is the altitude or height or distance between parallel sides.

Perimeter of Trapezium

The perimeter of a trapezium is calculated as the sum of all four sides.

Mathematically, Perimeter of Trapezium = (AB + BC + CD + AD)

 Where, “AB”, “BC”, “CD” and “AD” are respective sides of trapezium ABCD.

Solved Examples:

Q.1. Find the Length of Line that Joins the Midpoints of Non-parallel Sides of a Trapezium Whose Measure of Bases are 4 cm and 6 cm.

Ans. The length of line that joins the midpoints of non-parallel sides of the trapezium is given by the formula: EF = \[\frac {AB + CD}{2}\]

Here, AB = 4 cm and CD = 6 cm

So, EF =\[\frac {4 + 6}{2}\]=\[\frac {10}{2}\]= 5 cm  

Q.2. Find the Area of a Trapezium Whose Measure of Parallel Sides are 10 cm and 20 cm and Height 15 cm.

Ans. The area of a trapezium is given by = \[\frac {1}{2}\] x (Sum of Parallel Sides) x (Distance Between them).

Here, the measures of parallel sides are 10 cm and 20 cm.

So, their sum = 10 cm + 20 cm = 30 cm. and given that height is 15 cm.

Putting these values in above formula we get: 

\[\frac {1}{2}\] x (30 cm) x (15 cm) = 225 cm².

Q.3. Find the Perimeter of Trapezium ABCD Whose Side Measures are 10 cm, 12 cm, 14cm and 16 cm respectively.

Ans. The perimeter of Trapezium = Sum of its all four sides.

                                                          = 10 cm + 12 cm +14 cm + 16 cm 

                                                          = 52 cm.

Q.4. Do the Diagonals of Trapezium Bisect Each Other?

Ans. No, the diagonals of a trapezium might not bisect each other. If the diagonals are bisecting, the trapezium will be a parallelogram. So, every parallelogram is a trapezium but every trapezium might not be a parallelogram.

Q.5. What are the Different Types of a Trapezium?

Ans. There are three different types of trapezium as following:

1.    Isosceles trapezium

2.    Right trapezium

3.    Scalene trapezium 

Conclusion

Trapezium can be defined as the four-sided figure which is closed and has one pair of parallel sides opposite to each other and the other pair of sides are non-parallel. It is a quadrilateral, which is convex in shape which has 4 sides, 4 angles and 4 vertices. The sum of all the angles present interiorly is 360 degrees. The adjacent sides’ angle is 180 degrees and the diagonals of this structure bisect each other. The base of the trapezium is one of the parallel sides of the trapezium which are opposite to each other. On the other hand, the height of the trapezium is the perpendicular distance between two lines that are parallel.