Trapezium (Geometry)
Trapezium is a quadrilateral in which two sides are parallel and other two sides are non parallel.
It is one of the irregular quadrilateral which do have parallel opposite sides.
In fact the word “Trapezium” means “Little Table” in Greek.
Keyword for Trapezium
(a) one pair of parallel sides
(b) one pair of non-parallel sides
Structure of Trapezium
Trapezium is made up of following components
(a) Base
The parallel sides are known as Base.
(b) Leg
Sides which are non parallel are called Legs
(c) Height
Perpendicular distance between parallel lines is called height
(d) There are four angles in trapezium
Examples of Trapezium
Types of Trapezium
Trapezoids are classified on the basis of sides and angles.
(01) Scalene Trapezoid
Its a trapezium in which all angles and sides are of different measure
(02) Isosceles Trapezium
Its a trapezium in which the non parallel sides are equal
Given above is the trapezium ABCD in which AB & CD are parallel sides and AD & BC are non-parallel sides.
In Isosceles Trapezium, the non- parallel sides are equal
i.e. AD = BC
(03) Right Trapezium
A trapezium with at-least two right angles is called Right Trapezium
Given above is Right Trapezium ABCD in which, ∠A = ∠D = 90 degree
Properties of Trapezium
(01) One pair of sides are parallel to each other
In the above trapezium ABCD, side AB and CD are parallel
i.e. AB II CD
(02) The adjacent angles between parallel lines are supplementary
Given above is trapezium ABCD with line AB II CD
According to angle property of trapezium
∠1 + ∠ 2 = 180 and ∠3 + ∠ 4 = 180
Proof
Line AB and CD are parallel lines and AD is a transversal
So,
∠1 + ∠ 2 = 180 { Same side interior angle property }
Similarly, consider AB & CD as parallel lines and BC as transversal
∠3 + ∠ 4 = 180 {Same side interior angle property}
(03) Diagonals of Trapezium
Except Isosceles Trapezium, the diagonals do not bisect each other
(04) In Trapezium, diagonals divide each other proportionally
ABCD is a trapezium in which line AB is parallel to CD (AB II CD)
AC & BD are the two diagonals intersecting at O
According to trapezium property, diagonals divide each other proportionally.
\mathsf{\frac{DO}{OB} \ =\ \frac{CO}{OA}}
Proof
Taking triangle AOB & DOC
Considering AB & CD as parallel lines and BD as transversal
∠ 1 = ∠ 3 {Alternate Interior angle}
Considering AB II CD and AC as transversal
∠ 4 = ∠ 2 {Alternate Interior angle}
∠ 5 = ∠ 6 { Vertically opposite angles}
Hence by AAA similarity, ▵AOB & ▵DOC are similar.
In similar triangles, the ratio of corresponding sides are same
\mathsf{\frac{DO}{OC} \ =\ \frac{OB}{OA}}
On changing sides we get:
\mathsf{\frac{DO}{OB} \ =\ \frac{CO}{OA}}
Hence Proved
(05) Trapezium Diagonal Property
Sum of square of diagonals = Sum of square of non parallel side + twice of the product of parallel sides
Given is trapezium ABCD with parallel sides AB & CD.
AC & BD are the diagonals
According to Trapezium Diagonal property
\mathsf{AC^{2} +BD^{2} =\ AD^{2} +BC^{2} \ +\ 2\ AB\ .\ CD}
(06) Midsegment Theorem of Trapezium
The length of line joining the midpoint of non parallel side is equal to half of the sum of parallel sides
ABCD is a trapezium with AB & CD as parallel sides
E & F are the midpoints of non parallel sides AD & BC respectively
According to Midsegment Theorem
EF = 1/2 (AB + CD)
(07) Area formula for Trapezium
Area of Trapezium is given by formula:
\mathsf{Area\ =\ \frac{1}{2} \ ( sum\ of\ parallel\ sides) \ \times \ height}\\\ \\ \mathsf{Area\ =\ \frac{1}{2} \ ( AB+CD) \ \times \ h}
(08) Perimeter of Trapezium
Perimeter of any figure is the total length of the boundary.
In case of Trapezium, the perimeter is the sum of all sides
Perimeter = (a + b + c + d) cm
Questions on Trapezium
(01) ABCD is a trapezium in which AB || CD, AB = 16 cm and CD = 12 cm and distance between parallel sides is 10 cm. Find the area of trapezium
Given
AB = 16 cm
CD = 12 cm
Height (h) = 10 cm
We know that;
Area of Trapezium = (1/2) x (Sum of parallel sides) x height
Area = (1/2) ( AB + CD ) x h
Area = (1/2) (16 + 12) x 10 = 140 sq. cm
Hence 140 sq cm is the area of trapezium
(02) ABCD is a trapezium in which AB II CD. Diagonals intersect at point O.
Its given that AO = 3, OC = x – 3, OD = 3x – 19, OB = x – 5
Find the values of x
Using diagonal property of Trapezium
\mathsf{\frac{DO}{OB} \ =\ \frac{CO}{OA}}
\mathsf{\frac{3x-19}{x-5} \ =\ \frac{x-3}{3}}\\\ \\ \mathsf{3( 3x-19) \ =( x-5) \ ( x-3)}\\\ \\ \mathsf{9x\ -57\ =\ x^{2} -8x+15}\\\ \\ \mathsf{x^{2} -17x+72\ =\ 0}\
On solving the quadratic equation we get;
x = 9 and x = 8
Hence both 8 & 9 are the solution for x
Frequently Asked Questions – Trapezium
(01) How is Trapezium different from Parallelogram
In Parallelogram, opposite sides are equal and parallel, while in Trapezium, only one pair of sides are parallel.
(02) Are diagonals equal in Trapezium?
NO!!
Neither they are equal, neither they bisect each other
(03) In Trapezium, are opposite pair of angles equal?
There is no such property of trapezium.
Angle related property related to trapezium are:
(a) Alternate angles are supplementary
(b) Sum of all angles add to 360 degree
(04) Any other name of Trapezium?
In some countries, they are also called Trapezoid.
Summary of Trapezium
