Assertion (A): In rhombus ABCD, angle ABC = 120° and length of its each side is 20 cm. The length of diagonal BD = 20 cm.

Reason (R): In ΔAOB, cos 60° = $\dfrac{\text{OB}}{\text{20 cm}}$ and BD = 2 x OB

1. A is true, but R is false.

2. A is false, but R is true.

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

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

Given, angle ABC = 120° and length of its each side = 20 cm.

As we know that, each diagonal divides the angles at its endpoints into two equal parts.

The diagonal BD bisects ∠ABC and ∠ADC.

⇒ ∠ABD = ∠CBD = $\dfrac{120°}{2}$ = 60°

By formula,

cos θ = $\dfrac{\text{Base}}{\text{Hypotenuse}}$

We know that,

Diagonals of rhombus are perpendicular to each other.

Thus, triangle AOB is a right angle triangle.

In ΔAOB,

$\Rightarrow \text{cos B} = \dfrac{\text{OB}}{\text{AB}}\\[1em] \Rightarrow \text{cos 60°} = \dfrac{\text{OB}}{20}\\[1em] \Rightarrow \dfrac{1}{2} = \dfrac{\text{OB}}{20}\\[1em] \Rightarrow \text{OB} = \dfrac{20}{2}\\[1em] \Rightarrow \text{OB} = 10$

Since, diagonals of a rhombus bisect each other.

⇒ BD = 2 x OB

So, reason (R) is true.

⇒ BD = 2 x 10 cm = 20 cm

So, assertion (A) is true.

∴ Both A and R are true, and R is the correct reason for A.

Hence, option 3 is the correct option.