Two opposite angles of a rhombus are 60° each. If the length of each side of the rhombus is 8 cm, find the lengths of the diagonals of the rhombus.

Answer

To find the lengths of the diagonals of a rhombus where two opposite angles are 60° each and the length of each side is 8 cm, we can follow these steps:

### Step 1: Understand the properties of the rhombus
A rhombus has the following properties:
- All sides are equal in length.
- The diagonals bisect each other at right angles (90°).
- The opposite angles are equal.

Given that two opposite angles are 60°, the other two angles will also be 60° and 120°.

### Step 2: Draw the rhombus and label the angles
Let's label the rhombus as ABCD, where:
- Angle A = 60°
- Angle B = 120°
- Angle C = 60°
- Angle D = 120°

### Step 3: Draw the diagonals
Draw the diagonals AC and BD, which intersect at point M. Since the diagonals bisect each other at right angles, we have:
- Angle AMB = Angle CMD = 90°

### Step 4: Determine the angles at point M
Since the diagonals bisect the angles, we can find:
- Angle AMC = Angle BMD = 30° (because angle A = 60° and it is bisected)

### Step 5: Use trigonometry to find the lengths of the diagonals
In triangle AMC:
- AM is the height (perpendicular) from A to line BC.
- MC is half of diagonal AC.
- AC = 2 * MC.

Using the sine function:
\[ \sin(30°) = \frac{MC}{AC} \]
\[ \sin(30°) = \frac{1}{2} \implies \frac{MC}{8} = \frac{1}{2} \implies MC = 4 \text{ cm} \]

### Step 6: Find the length of diagonal AC
Since AC = 2 * MC:
\[ AC = 2 * 4 = 8 \text{ cm} \]

### Step 7: Find the length of diagonal BD using Pythagorean theorem
In triangle AMD:
- AM is the height (perpendicular) from A to line BD.
- MD is half of diagonal BD.
- BD = 2 * MD.

Using the Pythagorean theorem:
\[ AD^2 = AM^2 + MD^2 \]
Where AD = 8 cm (the side of the rhombus):
\[ 8^2 = AM^2 + MD^2 \]

We need to find AM. In triangle AMC:
\[ AM = AC \cdot \sin(60°) = 8 \cdot \frac{\sqrt{3}}{2} = 4\sqrt{3} \text{ cm} \]

Now substituting back:
\[ 64 = (4\sqrt{3})^2 + MD^2 \]
\[ 64 = 48 + MD^2 \]
\[ MD^2 = 64 - 48 = 16 \implies MD = 4 \text{ cm} \]

### Step 8: Find the length of diagonal BD
Since BD = 2 * MD:
\[ BD = 2 * 4 = 8 \text{ cm} \]

### Final Result
The lengths of the diagonals of the rhombus are:
- Diagonal AC = 8 cm
- Diagonal BD = 8√3 cm