Question

In the adjoining figure, a trapezium ABCD is shown in which AB || DC, AB = x and DC = y. If E and F are the mid-points of the sides AD and BC respectively, then ar (ABFE) : ar (EFCD) is:

Options

• x : y

• (3x + y) : (x + 3y)

• (x + 3y) : (3x + y)

• (2x + y) : (3x + y)

Answer 1

(3x + y) : (x + 3y)

Explanation:

Let the trapezium height be h.

The segment EF joining mid‑points of AD and BC is parallel to the bases and has length `(x + y)/2` and it lies midway so each smaller trapezoid has height `h/2`.

`"Area" (ABFE) = 1/2 xx (AB + EF) xx (h/2)` 

= `1/2 xx (x + (x + y)/2) xx (h/2)` 

= `((3x + y)h)/8`

`"Area" (EFCD) = 1/2 xx (EF + DC) xx (h/2)` 

= `((x + 3y)h)/8`

Thus, ar (ABFE) : ar (EFCD) = (3x + y) : (x + 3y).