The perimeter and the length of one of the diagonals of a rhombus are 26 cm and 5 cm respectively. Find the length of its other diagonal (in cm).

Options

A. 6
B. 12
C. 24
D. 18

Answer

To solve the problem, we need to find the length of the other diagonal of the rhombus given the perimeter and one diagonal.

### Step-by-Step Solution:

1. **Understand the properties of a rhombus**:
A rhombus has all four sides equal in length, and its diagonals bisect each other at right angles.

2. **Calculate the length of one side of the rhombus**:
The perimeter of the rhombus is given as 26 cm. Since a rhombus has 4 equal sides, we can find the length of one side (s) using the formula:
\[ \text{Perimeter} = 4s \]
Therefore,
\[ s = \frac{\text{Perimeter}}{4} = \frac{26 \text{ cm}}{4} = 6.5 \text{ cm} \]

3. **Use the diagonal lengths**:
Let the lengths of the diagonals be \(d_1\) and \(d_2\). We know that one diagonal \(d_1 = 5 \text{ cm}\). The diagonals bisect each other, so each half of the diagonals will be:
\[ \frac{d_1}{2} = \frac{5 \text{ cm}}{2} = 2.5 \text{ cm} \]
Let \(d_2\) be the length of the other diagonal. Thus, half of the other diagonal will be:
\[ \frac{d_2}{2} \]

4. **Apply the Pythagorean theorem**:
In the right triangle formed by the halves of the diagonals and the side of the rhombus, we can use the Pythagorean theorem:
\[ s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]
Substituting the known values:
\[ (6.5)^2 = (2.5)^2 + \left(\frac{d_2}{2}\right)^2 \]
\[ 42.25 = 6.25 + \left(\frac{d_2}{2}\right)^2 \]

5. **Solve for \(\left(\frac{d_2}{2}\right)^2\)**:
\[ \left(\frac{d_2}{2}\right)^2 = 42.25 - 6.25 = 36 \]

6. **Find \(\frac{d_2}{2}\)**:
\[ \frac{d_2}{2} = \sqrt{36} = 6 \text{ cm} \]

7. **Calculate \(d_2\)**:
\[ d_2 = 2 \times 6 = 12 \text{ cm} \]

### Conclusion:
The length of the other diagonal \(d_2\) is **12 cm**.

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