ABCD is a trapezium in which AB || CD and AB = 2 CD. If its diagonals interest each other at O, then

given: ABCD is a trapezium with AB||CD.......(1)
and AB = 2 CD......(2)

In the triangles AOB and COD;

∠DOC = ∠ BOA [vertically opposite angles are equal]
∠ CDO = ∠ ABO [alternate interior angles ]
∠ DCO = ∠ BAO

thus Δ AOB ≈ Δ COD.

by the similarity rule, the ratio of the areas of the similar triangles is the ratio of the square of corresponding sides.
therefore
area(Δ AOB) : area(Δ COD)=AB^2:CD^2
area(Δ AOB) : area(Δ COD)=(2CD)^2:CD^2
area(Δ AOB) : area(Δ COD)=4CD^2:CD^2
area(Δ AOB) : area(Δ COD) = 4 : 1.
So, option 9)4:1 is the right answer

1.angle DOC = BOA (vertically opposiete angles are equal)
2.angle CDO = ABO (alternate interior angle)
3. angle DCO = BAO
thus triangle AOB and Cod are similar
therefore by similarity rule
(area of triagle AOB)/(area of triangle COD) =(AB/CD)^2
=(2CD/CD)^2=4
=4:1

Triangle AOB and COD are similar Triangles
so, AB^2/CD^2 .------- (1)

Here, AB=2CD

Putting this in eq. 1,

We get, 4CD^2/CD^2

Hence ratio of area of triangle AOB and COD is.-- 4:1

Option 9)

AB||CD
and diagonals are intersecting at point o
----> angle AOB=angle COD(vertically opossite)
angle ABO=angle CDO(alternate angles)
similarly, angle BAO= angle DCO
Hence , Triangle AOB is congurent to Triangle COD
----->[ Area of triangle AOB/Area of triangle COD]
=(AB^2/CD^2)
=((2*CD)^2)/CD^2)
=4:1
Hence option 9 is correct :)

Since the parallel side AB=2CD ,the height considered from the intersection point on to the sides will also be in the ratio of 2:1
Hence The area AOB:COD=2*2:1*1=4:1

Since both triangles are similar to each other through AAA ,
=> area of triangle AOB = (AB*AB*h) / 2 * (AB + CD )
where h = height of trapezium
similarly ,
area of triangle COD = (CD * CD * h ) / 2 * (AB + CD)
now ,
ar AOB / ar COD = ((AB*AB*h) / 2 * (AB + CD ) ) / ((CD * CD * h ) / 2 * (AB + CD))
= (AB * AB * h) / (CD *CD * h)
= (AB*AB) / (CD * CD)
= (2*CD *2 * CD) / (CD * CD) ........since AB = 2*CD
= 4 / 1 .......................................ANS

both triangles are similar as all angles will be similar. And by the similarity rule>> ratio of area of similar triangle is equals to ratio of squares of corresponding sides so (AB)^2 /(CD)^2 = 4:1

4:1 is the answer.