The diagonals of a rhombus measure `16` cm and `30` cm. Find its perimeter

Options

A. `62cm`
B. `65cm`
C. `61cm`
D. `68 cm`

Explanation

To find the perimeter of a rhombus given its diagonals, we can follow these steps:

### Step 1: Identify the diagonals
Let the diagonals of the rhombus be \( AC \) and \( BD \). According to the problem, we have:
- \( AC = 16 \) cm
- \( BD = 30 \) cm

### Step 2: Find the half-lengths of the diagonals
Since the diagonals of a rhombus bisect each other at right angles, we can find the lengths of half of each diagonal:
- Half of diagonal \( AC \) (let's denote it as \( OA \)) = \( \frac{16}{2} = 8 \) cm
- Half of diagonal \( BD \) (let's denote it as \( OB \)) = \( \frac{30}{2} = 15 \) cm

### Step 3: Use the Pythagorean theorem
Now, we can use the Pythagorean theorem in triangle \( AOB \) to find the length of one side of the rhombus (which is equal for all sides). According to the Pythagorean theorem:
\[ AB^2 = OA^2 + OB^2 \]
Substituting the values we found:
\[ AB^2 = 8^2 + 15^2 \]
Calculating the squares:
\[ AB^2 = 64 + 225 \]
\[ AB^2 = 289 \]
Taking the square root to find \( AB \):
\[ AB = \sqrt{289} = 17 \text{ cm} \]

### Step 4: Calculate the perimeter of the rhombus
The perimeter \( P \) of a rhombus is given by the formula:
\[ P = 4 \times \text{side length} \]
Substituting the side length we found:
\[ P = 4 \times 17 = 68 \text{ cm} \]

### Final Answer:
The perimeter of the rhombus is \( 68 \) cm.

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