Statement 1: ABCD is a rhombus, its diagonal AC = 16 cm and diagonal BD = 12 cm, perimeter of rhombus = 64 cm.

Statement 2: OA = 8 cm, OB = 6 cm. Then, AB = 10 cm

And, perimeter of rhombus = 40 cm

1. Both the statements are true.

2. Both the statements are false.

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

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

Given, ABCD is a rhombus, diagonal AC = 16 cm and diagonal BD = 12 cm.

The diagonals of a rhombus bisect each other at right angles.

∴ ∠AOB = 90°

Each diagonal is divided into two equal segments.

∴ AC = 16 cm ⇒ AO = OC = $\dfrac{16}{2}$ = 8 cm

∴ BD = 12 cm ⇒ BO = OD = $\dfrac{12}{2}$ = 6 cm

According to Pythagoras theorem, in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

⇒ Hypotenuse² = Base² + Height²

In Δ AOB,

⇒ AB² = BO² + OA²

⇒ AB² = 6² + 8²

⇒ AB² = 36 + 64

⇒ AB² = 100

⇒ AB = $\sqrt{100}$

⇒ AB = 10 cm

According to formula, perimeter of rhombus = 4 x length of side

= 4 x 10

= 40 cm.

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

Hence, option 4 is the correct option.