#
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.

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 $\u2206$ 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 $\u2206$ 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 $\u2206$ 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 $\u2206$ 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 cm**

^{2}( Ans )Hope this information will clear your doubts about topic.

If you have any more doubts just ask here on the forum and our experts will try to help you out as soon as possible.

Regards

**
**