The length of a side of a square field is 4 m. What will be the altitude of the rhombus, if the area of the rhombus is equal to the area of a square field and one of its diagonal is 2 m ?

Answer

To solve the problem, we need to find the altitude of a rhombus given that the area of the rhombus is equal to the area of a square field, and one of its diagonals is 2 m.

### Step-by-Step Solution:

1. **Calculate the Area of the Square Field**:
- The length of a side of the square field is given as 4 m.
- The area of a square is calculated using the formula:
\[ \text{Area of square} = \text{side}^2 \]
- Substituting the value:
\[ \text{Area of square} = 4^2 = 16 \text{ m}^2 \]

**Hint**: Remember that the area of a square is found by squaring the length of one side.

2. **Set the Area of the Rhombus Equal to the Area of the Square**:
- Since the area of the rhombus is equal to the area of the square, we have:
\[ \text{Area of rhombus} = 16 \text{ m}^2 \]

**Hint**: The problem states that the areas are equal, so you can directly equate them.

3. **Use the Formula for the Area of a Rhombus**:
- The area of a rhombus can also be calculated using the formula:
\[ \text{Area of rhombus} = \frac{1}{2} \times d_1 \times d_2 \]
- Where \(d_1\) and \(d_2\) are the lengths of the diagonals. We know \(d_1 = 2 \text{ m}\) (given).

**Hint**: Remember that the area of a rhombus can be calculated using its diagonals.

4. **Substitute Known Values into the Area Formula**:
- We can substitute the known values into the area formula:
\[ 16 = \frac{1}{2} \times 2 \times d_2 \]
- Simplifying this gives:
\[ 16 = 1 \times d_2 \quad \Rightarrow \quad d_2 = 16 \text{ m} \]

**Hint**: Simplify the equation step by step to isolate the unknown diagonal.

5. **Relate Area to Altitude**:
- The area of a rhombus can also be expressed in terms of its side and altitude (h):
\[ \text{Area} = \text{side} \times \text{altitude} \]
- Let the side of the rhombus be \(s\). The altitude can be expressed as:
\[ h = \frac{\text{Area}}{s} \]

**Hint**: The altitude can be found by dividing the area by the length of a side.

6. **Calculate the Side of the Rhombus**:
- The side of the rhombus can be found using the diagonals:
\[ s = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2} \]
- Substituting the values:
\[ s = \sqrt{\left(\frac{2}{2}\right)^2 + \left(\frac{16}{2}\right)^2} = \sqrt{1^2 + 8^2} = \sqrt{1 + 64} = \sqrt{65} \]

**Hint**: Use the Pythagorean theorem to find the side length from the diagonals.

7. **Calculate the Altitude**:
- Now, substituting the area and side into the altitude formula:
\[ h = \frac{16}{\sqrt{65}} \]

**Hint**: Ensure that you substitute the correct values into the altitude formula.

### Final Answer:
The altitude of the rhombus is:
\[ h = \frac{16}{\sqrt{65}} \text{ m} \]