Assertion (A): Area of given rhombus = 10 cm x 10 cm = 100 cm²

Reason (R):

⇒ OA = $5\sqrt{3}$ cm
⇒ AC = 2 x $5\sqrt{3}$ cm = $10\sqrt{3}$ cm
⇒ OB = 5 cm
⇒ BD = 2 x 5 cm = 10 cm

Area of rhombus ABCD = AC x BD = 10 x 10 cm² = 100 cm²

1. A is true, R is false.
2. A is false, R is true.
3. Both A and R are true.
4. Both A and R are false.

Both A and R are false.

Explanation

We know that the diagonal of a rhombus bisect each other at right angles and also bisect the angle of the vertex.

⇒ OA = OC = $\dfrac{1}{2}$ AC

⇒ OB = OD = $\dfrac{1}{2}$ BD

∠ AOB = 90°

∠ OAB = 30°

In Δ AOB,

sin 30° = $\dfrac{OB}{AB}$

⇒ $\dfrac{1}{2} = \dfrac{OB}{10}$

⇒ OB $= \dfrac{10}{2}$

⇒ OB = 5 cm

And, cos 30° = $\dfrac{OA}{AB}$

⇒ $\dfrac{\sqrt3}{2} = \dfrac{OA}{10}$

⇒ OA $= \dfrac{10 \times \sqrt3}{2}$

⇒ OB = 5 $\sqrt3$ cm = 8.66 cm

So, AC = 2 x OA = 2 x 8.66 cm

= 17.32 cm

And, BD = 2 x OB = 2 x 5 cm

= 10 cm

Area of rhombus = $\dfrac{1}{2}$ x product of diagonals

= $\dfrac{1}{2}$ x 17.32 x 10 cm²

= 17.32 x 5 cm²

= 86.6 cm²

According to Assertion, Area of rhombus = 10 cm x 10 cm = 100 cm² (≠ 86.6 cm²).

∴ Assertion (A) is false.

Given, OA = $5\sqrt{3}$ cm

⇒ AC = 2 x $5\sqrt{3}$ cm = $10\sqrt{3}$ cm

And, OB = 5 cm

⇒ BD = 2 x 5 cm = 10 cm

Area of rhombus ABCD = AC x BD = $10\sqrt{3}$ x 10 cm² = 100 $\sqrt3$ cm²

∴ Reason (R) is false.

Hence, both Assertion (A) and Reason (R) are false.