In rhombus ABCD, diagonals AC and BD intersect each other at point O.
If cosine of angle CAB is 0.6 and OB = 8 cm, find the lengths of the side and the diagonals of the rhombus.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the properties of the rhombus
In a rhombus, the diagonals bisect each other at right angles. Therefore, if diagonals AC and BD intersect at point O, we have:
- AO = OC
- BO = OD
- ∠AOB = ∠COD = 90°

### Step 2: Use the given information
We are given:
- \( \cos(\angle CAB) = 0.6 \)
- \( OB = 8 \, \text{cm} \)

### Step 3: Identify the triangle to use
We will consider triangle OAB. In this triangle:
- \( OB \) is the adjacent side to angle CAB.
- \( OA \) is the opposite side to angle CAB.
- \( AB \) is the hypotenuse.

### Step 4: Use the cosine definition
From the definition of cosine:
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{OB}{AB} \]
Given \( \cos(\theta) = 0.6 \), we can express it as:
\[ 0.6 = \frac{OB}{AB} \]
Substituting \( OB = 8 \, \text{cm} \):
\[ 0.6 = \frac{8}{AB} \]

### Step 5: Solve for AB
Cross-multiplying gives:
\[ 0.6 \cdot AB = 8 \]
\[ AB = \frac{8}{0.6} = \frac{80}{6} = \frac{40}{3} \approx 13.33 \, \text{cm} \]

### Step 6: Find the lengths of the sides of the rhombus
Since all sides of a rhombus are equal, the length of each side \( AB = 10 \, \text{cm} \).

### Step 7: Use sine to find OA
We know that:
\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{OA}{AB} \]
Using the Pythagorean identity, if \( \cos(\theta) = 0.6 \), then:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
\[ \sin^2(\theta) + (0.6)^2 = 1 \]
\[ \sin^2(\theta) + 0.36 = 1 \]
\[ \sin^2(\theta) = 0.64 \implies \sin(\theta) = 0.8 \]

### Step 8: Solve for OA
Now substituting into the sine definition:
\[ 0.8 = \frac{OA}{AB} \]
Substituting \( AB = 10 \, \text{cm} \):
\[ 0.8 = \frac{OA}{10} \]
\[ OA = 0.8 \cdot 10 = 8 \, \text{cm} \]

### Step 9: Find the lengths of the diagonals
The lengths of the diagonals are:
- \( AC = 2 \cdot OA = 2 \cdot 6 = 12 \, \text{cm} \)
- \( BD = 2 \cdot OB = 2 \cdot 8 = 16 \, \text{cm} \)

### Final Results
- Length of each side of the rhombus: \( 10 \, \text{cm} \)
- Length of diagonal AC: \( 12 \, \text{cm} \)
- Length of diagonal BD: \( 16 \, \text{cm} \)