Question

the distance between parallel sides of a trapezium is 12 cm and the mid points of other sides is 18 cm. find the area of a trapezium

Answer 1

The distance between the parallel sides of the trapezium being 12 cm and the distance between the mid points of the other two sides being 18 cm, the area of the trapezium is 12x18 = 216 sq cm.

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Answer 2

Here , Diagonal BD intersect line EF at " O " .

And we assume O' is mid point of line BD .

In ∆ ABD , E and O' are mid points of AD and BD respectively , So from converse of mid point theorem we get

AB | | EO' ---- ( 1 )

And

In ∆ CDB , F and O' are mid points of BC and BD respectively , So from converse of mid point theorem we get

CD | | FO' , Given ABCD is a trapezium , SO AB | | CD , Then

AB | | FO' ---- ( 2 )

From equation 1 and 2 we can say that EO'F is a straight line , So O and O' coincide.

Therefore, O is mid point of BD

From equation 2 : AB | | OF , So

EF | | AB ( hence proved )

In ∆ ABD , E and O are mid points of AD and BD respectively , So from converse

OE = AB2 --- ( 3 )
In ∆ CDB , F and O are mid points of BC and BD respectively , So from converse of mid point theorem we get
OF = CD2 --- ( 4 )

Now we add equation 3 and 4 and get

OE + OF = AB2 + CD2



EF = AB + CD2 = Sum of parallel sides 2 --- ( 1 )

We know area of trapezium = Sum of parallel sides 2×Height

From equation ( 1 ) we get :

Area of given trapezium = EF × Height = 18 × 12 = 216 cm2 ( Ans )