The lengths of two diagonals of a rhombus are 12 cm and 16 cm. What is the side (in cm) of the rhombus?

Options

A. 10
B. 20
C. `10sqrt2`
D. `20sqrt2`

Answer

To find the side length of a rhombus given the lengths of its diagonals, we can use the relationship between the diagonals and the sides of the rhombus. Here’s how to solve the problem step by step:

### Step 1: Identify the lengths of the diagonals
Let the lengths of the diagonals be:
- \( D_1 = 12 \) cm
- \( D_2 = 16 \) cm

### Step 2: Use the formula relating the diagonals to the side of the rhombus
The formula that relates the lengths of the diagonals \( D_1 \) and \( D_2 \) to the side \( a \) of the rhombus is:
\[ D_1^2 + D_2^2 = 4a^2 \]

### Step 3: Substitute the values of the diagonals into the formula
Substituting the values of \( D_1 \) and \( D_2 \):
\[ 12^2 + 16^2 = 4a^2 \]

### Step 4: Calculate the squares of the diagonals
Calculating the squares:
\[ 12^2 = 144 \]
\[ 16^2 = 256 \]

### Step 5: Add the squares together
Now, add the results:
\[ 144 + 256 = 400 \]
So, we have:
\[ 400 = 4a^2 \]

### Step 6: Solve for \( a^2 \)
To find \( a^2 \), divide both sides by 4:
\[ a^2 = \frac{400}{4} = 100 \]

### Step 7: Take the square root to find \( a \)
Taking the square root of both sides gives:
\[ a = \sqrt{100} = 10 \text{ cm} \]

### Conclusion
The side length of the rhombus is \( 10 \) cm.