Understanding the Parallelogram Problem

The question asks for the perimeter of a parallelogram ABCD, given that one of its angles, \(\angle ABC\), is \(60^\circ\) and its area is \(7\sqrt{3}\) square units. We need to determine if the provided statements are sufficient to find this perimeter.

Parallelogram Properties and Area Formula

Let the lengths of the adjacent sides of the parallelogram be \(a\) and \(b\). In this case, let \(a = AB\) and \(b = BC\). The area of a parallelogram can be calculated using the formula:

Area = \(a \times b \times \sin(\theta)\)

where \(\theta\) is the angle between the sides \(a\) and \(b\). Here, \(\theta = \angle ABC = 60^\circ\). We are given that the Area = \(7\sqrt{3}\).

Substituting the values into the formula:

We know that \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\). So, the equation becomes:

To find the product of the sides \(a\) and \(b\), we can simplify this equation:

The perimeter of a parallelogram is given by the formula:

Perimeter = \(2 \times (a+b)\)

To find the perimeter, we need to determine the individual values of \(a\) and \(b\). Let's analyze the statements.

Statement-I Analysis

Statement-I says that the lengths of the sides AB and DA are prime numbers. Let \(a = AB\) and \(b = DA\). Note that adjacent sides of a parallelogram are usually denoted like AB and BC, or AB and AD. Let's assume the sides referred to are the adjacent sides whose product is 14. The statement implies that \(a\) and \(b\) are prime numbers, and their product \(a \times b = 14\).

We need to find pairs of prime numbers whose product is 14.

• The factors of 14 are: (1, 14) and (2, 7).
• Checking the pairs:
  • 1 is not a prime number.
  • 14 is not a prime number (it's divisible by 2 and 7).
  • 2 is a prime number.
  • 7 is a prime number.

The only pair of prime numbers whose product is 14 is (2, 7). Therefore, the lengths of the adjacent sides must be 2 units and 7 units.

Using these side lengths, the perimeter can be calculated:

Perimeter = \(2 \times (2 + 7) = 2 \times 9 = 18\) units.

Conclusion for Statement-I: The question can be answered using Statement-I alone.

Statement-II Analysis

Statement-II says that the lengths of the sides are natural numbers, and each is greater than 1 unit. Let the adjacent sides be \(a\) and \(b\). We know from the area calculation that \(a \times b = 14\).

We need to find pairs of natural numbers \((a, b)\) such that \(a > 1\), \(b > 1\), and \(a \times b = 14\).

The pairs of natural number factors of 14 are (1, 14) and (2, 7).

• Checking the pairs against the condition (\(a > 1\) and \(b > 1\)):
  • The pair (1, 14): Here, \(a=1\), which is not greater than 1. So, this pair is not valid under Statement-II.
  • The pair (2, 7): Here, \(a=2\) and \(b=7\). Both 2 and 7 are natural numbers greater than 1. This pair is valid.

Thus, the lengths of the adjacent sides must be 2 units and 7 units.

Using these side lengths, the perimeter can be calculated:

Perimeter = \(2 \times (2 + 7) = 2 \times 9 = 18\) units.

Conclusion for Statement-II: The question can be answered using Statement-II alone.

Overall Conclusion

Both Statement-I and Statement-II individually provide enough information to determine the lengths of the sides of the parallelogram and subsequently calculate its perimeter. Therefore, the question can be answered by using either Statement alone.