Find the perimeter of a rhombus whose diagonals are 30 cm and 34 cm.

Answer

To find the perimeter of a rhombus whose diagonals are 30 cm and 34 cm, we can follow these steps:

### Step 1: Understand the formula for the perimeter of a rhombus
The perimeter \( P \) of a rhombus can be calculated using the formula:
\[ P = 4 \times \text{side} \]
where "side" is the length of one side of the rhombus.

### Step 2: Find the length of the side using the diagonals
The length of a side of the rhombus can be found using the formula:
\[ \text{side} = \sqrt{\left(\frac{p}{2}\right)^2 + \left(\frac{q}{2}\right)^2} \]
where \( p \) and \( q \) are the lengths of the diagonals.

Given:
- \( p = 30 \, \text{cm} \)
- \( q = 34 \, \text{cm} \)

### Step 3: Calculate half of each diagonal
First, calculate half of each diagonal:
\[ \frac{p}{2} = \frac{30}{2} = 15 \, \text{cm} \]
\[ \frac{q}{2} = \frac{34}{2} = 17 \, \text{cm} \]

### Step 4: Substitute into the formula for the side
Now substitute these values into the formula for the side:
\[ \text{side} = \sqrt{(15)^2 + (17)^2} \]

### Step 5: Calculate the squares
Calculate the squares:
\[ (15)^2 = 225 \]
\[ (17)^2 = 289 \]

### Step 6: Add the squares
Now add the squares:
\[ 225 + 289 = 514 \]

### Step 7: Take the square root
Now take the square root to find the length of the side:
\[ \text{side} = \sqrt{514} \approx 22.67 \, \text{cm} \]

### Step 8: Calculate the perimeter
Now use the side length to find the perimeter:
\[ P = 4 \times 22.67 \approx 90.68 \, \text{cm} \]

### Final Answer
The perimeter of the rhombus is approximately \( 90.68 \, \text{cm} \).

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