If length of each side of a rhombus is 5 cm and length of one of its diagonal is 8 cm, find the length of other diagonal.

Options

A. A)5cm
B. B)6cm
C. C)7cm
D. D)8cm

Answer

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

### Step 1: Understand the properties of a rhombus
A rhombus has four equal sides and its diagonals bisect each other at right angles (90 degrees).

### Step 2: Draw the rhombus and label the points
Let the rhombus be ABCD, where AB = BC = CD = DA = 5 cm. Let the diagonals AC and BD intersect at point O. Given that the length of diagonal BD is 8 cm, we can find the lengths of segments BO and OD.

### Step 3: Calculate the lengths of segments BO and OD
Since the diagonals bisect each other, we have:
- BO = OD = 8 cm / 2 = 4 cm

### Step 4: Use the Pythagorean theorem in triangle AOB
In triangle AOB, we know:
- AB (the side of the rhombus) = 5 cm
- BO = 4 cm

We need to find OA (the half-length of diagonal AC).

Using the Pythagorean theorem:
\[ AB^2 = AO^2 + BO^2 \]
Substituting the known values:
\[ 5^2 = OA^2 + 4^2 \]
\[ 25 = OA^2 + 16 \]

### Step 5: Solve for OA
Rearranging the equation:
\[ OA^2 = 25 - 16 \]
\[ OA^2 = 9 \]
Taking the square root:
\[ OA = \sqrt{9} = 3 \text{ cm} \]

### Step 6: Find the length of diagonal AC
Since OA is half of diagonal AC, we can find AC:
\[ AC = 2 \times OA = 2 \times 3 = 6 \text{ cm} \]

### Final Answer
The length of the other diagonal AC is **6 cm**.

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