The lengths of two diagonals of a rhombus are 24cm and 32cm. What is the side (in cm) of the rhombus?

Options

A. 20
B. 40
C. `20 sqrt""2`
D. `40 sqrt""2`

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.

### Step-by-step Solution:

1. **Identify the lengths of the diagonals:**
Let \( d_1 = 24 \, \text{cm} \) and \( d_2 = 32 \, \text{cm} \).

2. **Use the formula relating the diagonals to the side of the rhombus:**
The formula is:
\[ a^2 = \frac{d_1^2 + d_2^2}{4} \]
where \( a \) is the length of a side of the rhombus.

3. **Calculate \( d_1^2 \) and \( d_2^2 \):**
\[ d_1^2 = 24^2 = 576 \]
\[ d_2^2 = 32^2 = 1024 \]

4. **Add the squares of the diagonals:**
\[ d_1^2 + d_2^2 = 576 + 1024 = 1600 \]

5. **Substitute into the formula:**
\[ a^2 = \frac{1600}{4} = 400 \]

6. **Take the square root to find \( a \):**
\[ a = \sqrt{400} = 20 \, \text{cm} \]

Thus, the length of each side of the rhombus is \( 20 \, \text{cm} \).