Question : Diagonals of a trapezium $ABCD$ with $AB \parallel CD$ intersect each other at the point $O$. If $AB = 2CD$, then the ratio of the areas of $\triangle AOB$ and $\triangle COD$ is:

Option 1: $4:1$

Option 2: $1:16$

Option 3: $1:4$

Option 4: $16:1$

Question : In trapezium $ABCD$, $AB \parallel CD$ and $AB = 2CD$. Its diagonals intersect at $O$. If the area of $\triangle AOB=84\;\mathrm{cm^2}$ then the area of $\triangle COD$ is equal to:

Option 1: 72 cm²

Option 2: 21 cm²

Option 3: 42 cm²

Option 4: 26 cm²

Question : $D$ and $E$ are points on the sides $AB$ and $AC$ respectively of $\triangle ABC$ such that $DE$ is parallel to $BC$ and $AD: DB = 4:5$, $CD$ and $BE$ intersect each other at $F$. Find the ratio of the areas of $\triangle DEF$ and $\triangle CBF$.

Option 1: $16:25$

Option 2: $16:81$

Option 3: $81:16$

Option 4: $4:9$

Question : ABCD is a quadrilateral in which BD and AC are diagonals. Then, which of the following is true:

Option 1: AB + BC + CD + DA < (AC + BD)

Option 2: AB + BC + CD + DA > (AC + BD)

Option 3: AB + BC + CD + DA = (AC + BD)

Option 4: AB + BC + CD + DA > 2(AC + BD)

Question : ABCD is a trapezium where AD$\parallel$ BC. The diagonal AC and BD intersect each other at the point O. If AO = 3, CO = $x-3$, BO = $3x-19$, and DO = $x-5$, the value of $x$ is:

Option 1: -8, 9

Option 2: 8, -9

Option 3: -8, -9

Option 4: 8, 9

Question : In a trapezium ABCD, AB and DC are parallel sides and $\angle ADC=90^\circ$. If AB = 15 cm, CD = 40 cm and diagonal AC = 41 cm, then the area of the trapezium ABCD is:

Option 1: 245 cm²

Option 2: 240 cm²

Option 3: 247.5 cm²

Option 4: 250 cm²