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 with diagonals measuring 16 cm and 30 cm, we can follow these steps:

### Step 1: Understand the properties of the rhombus
The diagonals of a rhombus bisect each other at right angles. This means that each diagonal is divided into two equal parts.

### Step 2: Calculate the half-lengths of the diagonals
Given:
- Diagonal AC = 16 cm
- Diagonal BD = 30 cm

Since the diagonals bisect each other:
- AO = OC = AC/2 = 16 cm / 2 = 8 cm
- OB = OD = BD/2 = 30 cm / 2 = 15 cm

### Step 3: Use the Pythagorean theorem to find the length of one side of the rhombus
In triangle AOB, we have:
- AO = 8 cm (one half of diagonal AC)
- OB = 15 cm (one half of diagonal BD)

According to the Pythagorean theorem:
\[ AB^2 = AO^2 + OB^2 \]
\[ AB^2 = 8^2 + 15^2 \]
\[ AB^2 = 64 + 225 \]
\[ AB^2 = 289 \]

Now, take the square root of both sides to find AB:
\[ AB = \sqrt{289} = 17 \text{ cm} \]

### Step 4: Calculate the perimeter of the rhombus
Since all sides of a rhombus are equal, the perimeter (P) can be calculated as:
\[ P = 4 \times AB \]
\[ P = 4 \times 17 \text{ cm} \]
\[ P = 68 \text{ cm} \]

### Final Answer
The perimeter of the rhombus is **68 cm**.

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