The lengths of the diagonals of a rhombus are 60 cm and 80 cm. Find the perpendicular distance between the opposite sides of rhombus.

Answer

To find the perpendicular distance between the opposite sides of a rhombus with diagonals measuring 60 cm and 80 cm, 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 lengths of the halves of the diagonals
- The length of the first diagonal is 60 cm, so each half is:
\[ \frac{60}{2} = 30 \text{ cm} \]
- The length of the second diagonal is 80 cm, so each half is:
\[ \frac{80}{2} = 40 \text{ cm} \]

### Step 3: Use the Pythagorean theorem to find the length of a side of the rhombus
Let \( A \) be the length of a side of the rhombus. Using the halves of the diagonals as the legs of a right triangle, we can apply the Pythagorean theorem:
\[ A^2 = 30^2 + 40^2 \]
Calculating the squares:
\[ A^2 = 900 + 1600 = 2500 \]
Taking the square root:
\[ A = \sqrt{2500} = 50 \text{ cm} \]

### Step 4: Calculate the area of the rhombus
The area \( A_r \) of a rhombus can also be calculated using the lengths of the diagonals:
\[ A_r = \frac{1}{2} \times d_1 \times d_2 \]
Substituting the values of the diagonals:
\[ A_r = \frac{1}{2} \times 60 \times 80 = \frac{4800}{2} = 2400 \text{ cm}^2 \]

### Step 5: Use the area to find the perpendicular distance between opposite sides
The area of a rhombus can also be expressed as:
\[ A_r = \text{side} \times \text{perpendicular distance} \]
Let \( D \) be the perpendicular distance. We have:
\[ 2400 = 50 \times D \]
Solving for \( D \):
\[ D = \frac{2400}{50} = 48 \text{ cm} \]

### Final Answer
The perpendicular distance between the opposite sides of the rhombus is **48 cm**.

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