The distance between parallel sides of a trapezium is 12 cm and the distance between mid-points of other sides is 18 cm. Find the area of the trapezium.
Hint: Let ABCD be the given trapezium in which AB ||  DC. Let E and F be mid-points of sides AD and BC respectively, then EF = 18 cm.
Area of trap. ABCD = $\frac{1}{2}$ (AB + DC) × height = EF × height.

Dear Student,

Please find below the solution to the asked query:

Given : The distance between parallel sides of a trapezium is 12 cm , So Height of trapezium  =  12 cm 

And the distance between mid - points of other sides is 18 cm. , Let ABCD be the given trapezium in which AB ||  DC. Let E and F be mid-points of sides AD and BC respectively, then EF = 18 cm.

We form our diagram from given information , As :

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 of mid point theorem we get

OE  = $\frac{\mathrm{AB}}{2}$                                        --- ( 3 )
In $∆$ CDB ,  F and O are mid points of BC and BD respectively , So from converse of mid point theorem we get
OF  = $\frac{\mathrm{CD}}{2}$                                        --- ( 4 )

Now we add equation 3 and 4 and get

OE  + OF = $\frac{\mathrm{AB}}{2}$ + $\frac{\mathrm{CD}}{2}$

EF  = $\frac{\mathrm{AB} + \mathrm{CD}}{2} = \frac{\mathrm{Sum} \mathrm{of} \mathrm{parallel} \mathrm{sides} }{2}$                        --- ( 1 )

We know area of trapezium  = $\frac{\mathrm{Sum} \mathrm{of} \mathrm{parallel} \mathrm{sides} }{2}\times \mathrm{Height}$

From equation ( 1 ) we get :

Area of given trapezium = EF $\times$ Height  =  18  $\times$ 12 =  216  cm²                                  ( Ans )

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