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To solve this problem, let's analyze the information given.

### Part 1: Finding the Perimeter of the Parallelogram

1. **Given Data**:
   - Diagonals of the parallelogram: \( d_1 = 16 \) units and \( d_2 = 30 \) units.
   - Length of one side of the parallelogram: \( a = 17 \) units.

2. **Goal**:
   - Find the perimeter of the parallelogram.

3. **Understanding the Shape**:
   - In a parallelogram, the opposite sides are equal in length. Therefore, if one side is 17 units, the opposite side will also be 17 units.
   - Perimeter of a parallelogram is given by \( P = 2(a + b) \), where \( a \) and \( b \) are the lengths of the two adjacent sides.

4. **More Specific Name for the Quadrilateral**:
   - Since the diagonals have different lengths, this is a generic parallelogram and not a rhombus (which would require equal diagonals).

#### Solution Outline
To find \( b \), we need to use additional information, such as the relationship involving the diagonals and the Law of Cosines, to get the angle between the sides or verify if any additional properties apply.

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### Part 2: Alternate Case with Side Length \( \sqrt{161} \)

1. **Updated Side Length**:
   - Now, the side of the parallelogram is \( a = \sqrt{161} \) units.

2. **Law of Cosines**:
   - To apply the Law of Cosines in this scenario, we need to calculate the angle between the sides, which will help determine the second side length \( b \) based on the updated configuration.

Would you like a more detailed solution on finding \( b \) using the Law of Cosines, or any further assistance with the setup?

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#### Related Questions
1. How does the Law of Cosines apply to finding the length of a side in a parallelogram?
2. What additional information would we need to identify this as a rhombus?
3. How can we use diagonal lengths to find angles in a parallelogram?
4. What are the conditions for a quadrilateral to be a rhombus?
5. How would the solution change if both diagonals were equal?

#### Tip
In a parallelogram, knowing both diagonals and one side can often help determine if special properties (like those of a rectangle or rhombus) apply.

The length of the diagonals of a parallelogram are 16 units and 30 units. If the length of one side of the parallelogram is 17 units, what is the perimeter of the parallelogram? Is there a more specific name for this quadrilateral?

Also - Find the perimeter if the length of one side of the parallelogram is sqrt(161) instead of 17. You may need the Law of Cosines to solve this problem.

Solve a problem involving a parallelogram with diagonals of 16 and 30 units and one side of 17 units. Learn how to calculate the perimeter and explore whether the shape has a more specific classification. Includes a variation requiring the Law of Cosines for a side length of sqrt(161). Suitable for students in Grades 10-12.

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