*ABCD* is a trapezium with parallel sides *AB* =*a* and *DC* = *b*. If *E* and *F* are mid-points of non-parallel sides *AD* and *BC* respectively, then the ratio of areas of quadrilaterals *ABFE*and *EFCD* is

#### Options

*a*:*b*(

*a*+ 3*b*): (3*a*+*b*)(3

*a*+*b*) : (*a*+ 3*b*)(2

*a*+*b*) : (3*a*+*b*)

#### Solution

**Given: **(1) ABCD is a trapezium, with parallel sides AB and DC

(2) AB = a cm

(3) DC = b cm

(4) E is the midpoint of non parallel sides AD.

(5) G is the midpoint of non parallel sides BC.

**To find:** Ratio of the area of the Quadrilaterals ABFE and EFCD.

**Calculation: **We know that, ‘**Area of a trapezium is half the product of its height and the sum of the parallel sides.’**

Since, E and F are mid points of AD and BC respectively, so h_{1} = h_{2}

Area of trapezium ABFE

Area of trapezium ABFE`=1/2 (a + x) h_1 = 1/2 (a + x)h`

Area of trapezium EFCD = `1/2 (b+x)h_2 = 1/2 (b+ x) h`

Area of trapezium ABCD `= 1/2 (a+ b) (h_1 + h_2 ) = (a+b) h`

Now, Area (trap ABCD) = area (trap EFCD) + Area (ABFE)

Therefore,

(a + b) h = `1/2 ( a + x) h + 1/2 (b+x) h`

`⇒ A +b = (a+b)/2 +x`

`⇒x = (a+b)/2`

Thus,

`(Area (ABFE))/(Area (EFCD)) = (a+x)/(b +x) = (a +(a+b)/2)/(b + (a+b)/2)`

`⇒ (Area (ABFE))/(Area (EFCD)) = (3a +b)/(a + 3b)`