The perimeter of a rhombus is 40 cm. If the length of one of its diagonals be 12 m, then the length of the other diagonal is

Options

A. 14 cm
B. 15 cm
C. 16 cm
D. 12 cm

Answer

To find the length of the other diagonal of a rhombus when given the perimeter and one diagonal, we can follow these steps:

### 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. **Given data**:
- Perimeter (P) of the rhombus = 40 cm
- Length of one diagonal (d1) = 12 cm

3. **Calculate the length of one side of the rhombus**:
- The formula for the perimeter of a rhombus is:
\[ P = 4 \times \text{side} \]
- Rearranging this gives:
\[ \text{side} = \frac{P}{4} = \frac{40 \text{ cm}}{4} = 10 \text{ cm} \]

4. **Determine the lengths of the diagonals**:
- Let the length of the second diagonal be \(d2\).
- Since the diagonals bisect each other at right angles, we can form two right triangles using half of each diagonal.
- Half of the first diagonal \(d1\) is:
\[ \frac{d1}{2} = \frac{12 \text{ cm}}{2} = 6 \text{ cm} \]
- Half of the second diagonal \(d2\) is:
\[ \frac{d2}{2} \]

5. **Apply the Pythagorean theorem**:
- In one of the right triangles formed, we have:
\[ (\text{half of } d1)^2 + (\text{half of } d2)^2 = (\text{side})^2 \]
- Substituting the known values:
\[ 6^2 + \left(\frac{d2}{2}\right)^2 = 10^2 \]
- This simplifies to:
\[ 36 + \left(\frac{d2}{2}\right)^2 = 100 \]

6. **Solve for \(\frac{d2}{2}\)**:
- Rearranging gives:
\[ \left(\frac{d2}{2}\right)^2 = 100 - 36 = 64 \]
- Taking the square root:
\[ \frac{d2}{2} = 8 \]

7. **Calculate \(d2\)**:
- Therefore, multiplying by 2 gives:
\[ d2 = 2 \times 8 = 16 \text{ cm} \]

### Final Answer:
The length of the other diagonal is **16 cm**.

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