The side of a rhombus is 13 cm. if one if the diagonals is 24 cm, find the length of the other diagonal.

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 two diagonals that bisect each other at right angles. This means that each diagonal divides the rhombus into four right-angled triangles.

### Step 2: Identify the given values
- The length of one side of the rhombus (AD) = 13 cm
- The length of one diagonal (AC) = 24 cm

### Step 3: Calculate half of the given diagonal
Since the diagonals bisect each other, we can find half of diagonal AC:
- Half of diagonal AC (AO) = 24 cm / 2 = 12 cm

### Step 4: Set up the right triangle
We can consider triangle AOD, where:
- AO = 12 cm (half of diagonal AC)
- AD = 13 cm (side of the rhombus)
- OD = ? (half of the other diagonal, which we will find)

### Step 5: Apply the Pythagorean theorem
In triangle AOD, we can use the Pythagorean theorem:
\[ AD^2 = AO^2 + OD^2 \]
Substituting the known values:
\[ 13^2 = 12^2 + OD^2 \]

### Step 6: Calculate the squares
Calculating the squares:
- \( 13^2 = 169 \)
- \( 12^2 = 144 \)

### Step 7: Solve for OD
Now we can substitute these values into the equation:
\[ 169 = 144 + OD^2 \]
Subtract 144 from both sides:
\[ OD^2 = 169 - 144 \]
\[ OD^2 = 25 \]

### Step 8: Find OD
Taking the square root of both sides gives us:
\[ OD = \sqrt{25} = 5 \text{ cm} \]

### Step 9: Calculate the length of the other diagonal
Since OD is half of the other diagonal (DB), we can find DB:
\[ DB = 2 \times OD = 2 \times 5 = 10 \text{ cm} \]

### Final Answer
The length of the other diagonal is **10 cm**.

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