Question

ABCD is a tapezium 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 quadrialaterals ABFE and EFCD is

• A. a : b
  
• B. (a + 3b) : (3a+b)
• C. (3a + b) : (a + 3b)
  
• D. (2a+ b ) : (3a + b)
  

Answer 1

The correct option is C

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

In quadrilateral ABCD, E and F are the mid points of AD and BC
$AB=a,CD=b$

$\therefore EF=\frac{a+b}{2}$
Let h be the height of trapezium ABCD, then
Altitude of quadrilateral ABFE = altitude of quadrilateral $EFCD=\frac{h}{2}$
Now area of trap. $ABFE=\frac{1}{2}\left(sumofparallelsides\right)\times altitude$
$=\frac{1}{2}(a+\frac{a+b}{2})\times \frac{h}{2}=\frac{h}{4}\left(\frac{2a+a+b}{2}\right)=\frac{h}{8}(3a+b)$
and area of trap. $EFCD=\frac{1}{2}(b+\frac{a+b}{2})\times \frac{h}{2}=\frac{h}{4}\left(\frac{2b+a+b}{2}\right)=\frac{h}{8}(a+3b)$
Now ratio between the area of these trapeziums
$=\frac{h}{8}(3a+b):\frac{h}{8}(a+3b)\Rightarrow 3a+b:a+3b$