If a rhombus of side 10 cm has two angles of `60^(@)` each. Then the length of diagonals ( in cm) are

Options

A. `10 sqrt3, 10`
B. `10 sqrt2, 10`
C. `5,10`
D. None of these

Answer

To find the lengths of the diagonals of a rhombus with a side length of 10 cm and two angles of 60 degrees each, we can follow these steps:

### Step 1: Understand the properties of the rhombus
A rhombus has all sides equal, and its diagonals bisect each other at right angles. Given that the side length is 10 cm and two angles are 60 degrees, we can denote the vertices of the rhombus as A, B, C, and D.

### Step 2: Identify the angles
Since two angles are given as 60 degrees, the remaining two angles will be 120 degrees each (because the sum of angles in a quadrilateral is 360 degrees).

### Step 3: Draw the diagonals
Let the diagonals AC and BD intersect at point O. The diagonals bisect each other at right angles, meaning that AO = OC and BO = OD.

### Step 4: Analyze triangle OBC
We will focus on triangle OBC. Here, angle BOC is 90 degrees (since diagonals bisect at right angles), angle OBC is 30 degrees (half of angle ABC which is 60 degrees), and OB is the hypotenuse of triangle OBC.

### Step 5: Use trigonometric ratios to find OB
Using the sine function:
\[ \sin(30^\circ) = \frac{OB}{BC} \]
Here, BC (the side of the rhombus) is 10 cm, and \(\sin(30^\circ) = \frac{1}{2}\):
\[ \frac{1}{2} = \frac{OB}{10} \]
Thus, we find:
\[ OB = 10 \times \frac{1}{2} = 5 \text{ cm} \]

### Step 6: Use the cosine function to find OC
Using the cosine function:
\[ \cos(30^\circ) = \frac{OC}{BC} \]
Here, \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\):
\[ \frac{\sqrt{3}}{2} = \frac{OC}{10} \]
Thus, we find:
\[ OC = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \text{ cm} \]

### Step 7: Find the lengths of the diagonals
Since diagonals bisect each other:
- The length of diagonal AC is:
\[ AC = AO + OC = 5\sqrt{3} + 5\sqrt{3} = 10\sqrt{3} \text{ cm} \]
- The length of diagonal BD is:
\[ BD = OB + OD = 5 + 5 = 10 \text{ cm} \]

### Final Answer
The lengths of the diagonals are:
- AC = \(10\sqrt{3}\) cm
- BD = \(10\) cm