The perimeter of a rhombus is 52 metres while its longer diagonal is 24 metres .Its Other diagonal is -----

Options

A. 5 metres
B. 10 metres
C. 20 metres
D. 28 metres

Answer

To find the length of the other diagonal of the rhombus, we can follow these steps:

### Step 1: Understand the properties of the rhombus
A rhombus has four equal sides and its diagonals bisect each other at right angles. The perimeter of the rhombus is given as 52 meters.

### Step 2: Calculate the length of one side of the rhombus
The perimeter \( P \) of a rhombus is given by the formula:
\[ P = 4a \]
where \( a \) is the length of one side. Given that the perimeter is 52 meters, we can find \( a \):
\[ 4a = 52 \implies a = \frac{52}{4} = 13 \text{ meters} \]

### Step 3: Use the diagonals to find the other diagonal
Let the lengths of the diagonals be \( d_1 \) and \( d_2 \). We know that the longer diagonal \( d_1 \) is 24 meters, and we need to find the other diagonal \( d_2 \).

In a rhombus, the relationship between the sides and the diagonals is given by the formula:
\[ a^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]
Substituting the known values:
\[ 13^2 = \left(\frac{24}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 \]
\[ 169 = 12^2 + \left(\frac{d_2}{2}\right)^2 \]
\[ 169 = 144 + \left(\frac{d_2}{2}\right)^2 \]

### Step 4: Solve for \( d_2 \)
Rearranging the equation:
\[ \left(\frac{d_2}{2}\right)^2 = 169 - 144 \]
\[ \left(\frac{d_2}{2}\right)^2 = 25 \]
Taking the square root of both sides:
\[ \frac{d_2}{2} = 5 \implies d_2 = 10 \text{ meters} \]

### Conclusion
The length of the other diagonal \( d_2 \) is 10 meters.

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