If each side of a rhombus is 10 cm and one of its diagonals is 16 cm , then the length of the other diagonal is k cm . Find k .

Answer

To find the length of the other diagonal \( k \) of the rhombus, we can follow these steps:

### Step 1: Understand the properties of the rhombus
A rhombus has the following properties:
- All sides are equal.
- The diagonals bisect each other at right angles.

Given:
- Each side of the rhombus \( AB = BC = CD = DA = 10 \) cm.
- One diagonal \( BD = 16 \) cm.

### Step 2: Find the lengths of the segments created by the diagonals
Let \( O \) be the point where the diagonals intersect. Since the diagonals bisect each other, we have:
- \( BO = OD = \frac{BD}{2} = \frac{16}{2} = 8 \) cm.

### Step 3: Use the Pythagorean theorem in triangle \( AOB \)
In triangle \( AOB \):
- \( AB \) is the hypotenuse (10 cm),
- \( OB = 8 \) cm,
- Let \( OA = x \) cm.

According to the Pythagorean theorem:
\[ AB^2 = AO^2 + OB^2 \]
Substituting the known values:
\[ 10^2 = x^2 + 8^2 \]
\[ 100 = x^2 + 64 \]

### Step 4: Solve for \( x \)
Rearranging the equation:
\[ x^2 = 100 - 64 \]
\[ x^2 = 36 \]
Taking the square root:
\[ x = \sqrt{36} = 6 \text{ cm} \]

### Step 5: Find the length of diagonal \( AC \)
Since \( AC \) is twice the length of \( OA \):
\[ AC = 2 \times OA = 2 \times 6 = 12 \text{ cm} \]

### Conclusion
Thus, the length of the other diagonal \( k \) is:
\[ k = 12 \text{ cm} \]

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