The diagonals of a rhombus are 32 cm and 60 cm. What is the perimeter of the rhombus?

Options

A. 68
B. 132
C. 136
D. 236

Answer

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

### Step 1: Understand the properties of a rhombus
A rhombus has two diagonals that bisect each other at right angles (90 degrees). This means that each diagonal divides the rhombus into four right-angled triangles.

### Step 2: Identify the lengths of the diagonals
The lengths of the diagonals are given as:
- Diagonal 1 (d1) = 32 cm
- Diagonal 2 (d2) = 60 cm

### Step 3: Calculate the half-lengths of the diagonals
Since the diagonals bisect each other, we can find the lengths of half of each diagonal:
- Half of diagonal 1 = d1/2 = 32 cm / 2 = 16 cm
- Half of diagonal 2 = d2/2 = 60 cm / 2 = 30 cm

### Step 4: Use the Pythagorean theorem to find the side length of the rhombus
In one of the right-angled triangles formed by the diagonals, we can use the Pythagorean theorem to find the length of a side (s) of the rhombus:
\[ s^2 = (d1/2)^2 + (d2/2)^2 \]
\[ s^2 = 16^2 + 30^2 \]
\[ s^2 = 256 + 900 \]
\[ s^2 = 1156 \]
Now, take the square root to find s:
\[ s = \sqrt{1156} = 34 \text{ cm} \]

### Step 5: Calculate the perimeter of the rhombus
The perimeter (P) of a rhombus is given by the formula:
\[ P = 4 \times s \]
Substituting the value of s:
\[ P = 4 \times 34 = 136 \text{ cm} \]

### Final Answer
The perimeter of the rhombus is **136 cm**.

---