In triangle ABC, AD : DB = 2 : 3, DE is parallel to BC.

Assertion (A) : $\dfrac{\text{DE}}{\text{BC}} = \dfrac{\text{AD}}{\text{BD}} = \dfrac{2}{3}$.

Reason (R) : $\dfrac{\text{DE}}{\text{BC}} = \dfrac{\text{AD}}{\text{AB}} = \dfrac{2}{5}$.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true and R is correct reason for A.

4. Both A and R are true and R is incorrect reason for A.

In triangle ODQ, ∠Q = ∠BPO = 90° AB = 2 x OA, BC = 3 x OA and CD = 4 x OA.

Assertion (A) : $\dfrac{\text{Δ OBP}}{\text{Trapezium BDQP}} = \dfrac{9}{100 - 9}$.

Reason (R) : Δ OBP - ODQ and $\dfrac{\text{Δ OBP}}{\text{Δ ODQ}} = \dfrac{(3a)^2}{(10a)^2}$.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true and R is correct reason for A.

4. Both A and R are true and R is incorrect reason for A.

Area of triangle ADE = 9 cm² and area of trapezium DBCE = 16 cm².

Statement (1) : $\dfrac{\text{DE}}{\text{BC}} = \dfrac{3}{4}$.

Statement (2) : $\dfrac{\text{ΔADE}}{\text{ΔABC}} = \dfrac{9}{25} \Rightarrow \dfrac{\text{DE}}{\text{BC}} = \dfrac{3}{5}$.

1. Both the statement are true.

2. Both the statement are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

In the following figure, XY is parallel to BC, AX = 9 cm, XB = 4.5 cm and BC = 18 cm.

Find:

(i) $\dfrac{AY}{YC}$

(ii) $\dfrac{YC}{AC}$

(iii) XY