Question

ABCD is a trapezium with parallel sides AB = a cm and DC = b cm (Figure). E and F are the mid-points of the non-parallel sides. The ratio of ar (ABFE) and ar (EFCD) is ______.

Options

• a : b

• (3a + b) : (a + 3b)

• (a + 3b) : (3a + b)

• (2a + b) : (3a + b)

Answer 1

ABCD is a trapezium with parallel sides AB = a cm and DC = b cm (Figure). E and F are the mid-points of the non-parallel sides. The ratio of ar (ABFE) and ar (EFCD) is (3a + b) : (a + 3b).
Explanation:

Given, AB = a cm, DC = b cm and AB || DC.

Also, E and F are mid-points of AD and BC, respectively.

So, distance between CD, EF and AB, EF will be same say h.

Join BD which intersect EF at M.

Now, in ΔABD, E is the mid-point of AD and EM || AB

So, M is the mid-point of BD

And EM = `1/2`AB   [By mid-point theorem]  ...(i)

Similarly in ΔCBD, MF = `1/2`CD  ...(ii)

On adding equations (i) and (ii), we get

EM + MF = `1/2` AB + `1/2` CD

⇒ EF = `1/2`(AB + CD) = `1/2`(a + b)

Now, area of trapezium ABFE

 = `1/2`(sum of parallel sides) × (distance between parallel sides)

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

= `1/4(3a + b)h`

Now, area of trapezium EFCD

= `1/2[b + 1/2(a + b)] xx h`

= `1/4(3b + a)h`

∴ Required ratio = `"Area of ABFE"/"Area of EFCD"`

= `(1/4(3a + b)h)/(1/4(3b + a)h)`

= `((3a + b))/((a + 3b))` or (3a + b) : (a + 3b)