If one of the diagonals of a rhombus is equal to its side, then the diagonals of the rhombus are in the ratio

Options

A. `sqrt(3):1`
B. `sqrt(2):1`
C. `3:1`
D. `2:1`

Answer

To solve the problem, we need to find the ratio of the diagonals of a rhombus when one of the diagonals is equal to the length of a side of the rhombus.

### Step-by-Step Solution:

1. **Understand the Properties of a Rhombus**:
A rhombus has four equal sides and its diagonals bisect each other at right angles.

2. **Let the Side Length be 'a'**:
Assume the length of each side of the rhombus is 'a'.

3. **Identify the Diagonal Equal to the Side**:
According to the problem, let one of the diagonals (let's call it d1) be equal to the side of the rhombus. Thus, \( d_1 = a \).

4. **Use the Properties of Right Triangles**:
The diagonals of the rhombus bisect each other at right angles. Therefore, each half of diagonal d1 will be \( \frac{d_1}{2} = \frac{a}{2} \).

5. **Apply the Pythagorean Theorem**:
In the right triangle formed by half of the diagonals and the side of the rhombus, we can denote the other diagonal as d2. The relationship can be expressed as:
\[ \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = a^2 \]
Plugging in \( d_1 = a \):
\[ \left(\frac{a}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = a^2 \]

6. **Simplify the Equation**:
\[ \frac{a^2}{4} + \frac{d_2^2}{4} = a^2 \]
Multiply through by 4 to eliminate the fractions:
\[ a^2 + d_2^2 = 4a^2 \]
Rearranging gives:
\[ d_2^2 = 4a^2 - a^2 = 3a^2 \]

7. **Find the Length of the Second Diagonal**:
Taking the square root of both sides:
\[ d_2 = \sqrt{3}a \]

8. **Determine the Ratio of the Diagonals**:
Now we have:
- \( d_1 = a \)
- \( d_2 = \sqrt{3}a \)

The ratio of the diagonals \( d_1 : d_2 \) is:
\[ \frac{d_1}{d_2} = \frac{a}{\sqrt{3}a} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} : 1 \]

### Final Answer:
The diagonals of the rhombus are in the ratio \( 1 : \sqrt{3} \).