If the length of the side PQ of the rhombus PQRS is 6 cm and `anglePQR=120^(@)`, then the find of QS, in cm, is

Options

A. A)3
B. B)5
C. C)4
D. D)6

Answer

To solve the problem of finding the length of diagonal QS in the rhombus PQRS, we can follow these steps:

### Step 1: Understand the properties of a rhombus
A rhombus has all four sides equal, and the diagonals bisect each other at right angles. Given that side PQ = 6 cm, we know that all sides of the rhombus (PQ, QR, RS, SP) are also 6 cm.

**Hint:** Remember that in a rhombus, all sides are equal, and the diagonals bisect the angles.

### Step 2: Draw the rhombus and label the angles
Draw rhombus PQRS and label the angles. We know that angle PQR = 120°. Since the diagonals bisect the angles, angle PQS will be half of angle PQR.

**Hint:** Visualizing the rhombus helps in understanding the angle relationships.

### Step 3: Calculate the angles at point Q
Since angle PQR = 120°, the angle PQS will be:
\[ \text{Angle PQS} = \frac{120°}{2} = 60° \]
Thus, angle PQS = 60° and angle QRP = 60°.

**Hint:** Use the property of angle bisectors to find the angles in the triangles formed.

### Step 4: Analyze triangle PSQ
In triangle PSQ, we have:
- Angle PQS = 60°
- Angle QPS = 60°
- Therefore, angle PSQ = 180° - (60° + 60°) = 60°.

This means triangle PSQ is an equilateral triangle.

**Hint:** Recognizing that all angles are equal helps identify the type of triangle.

### Step 5: Determine the length of QS
Since triangle PSQ is equilateral, all sides are equal. Therefore:
\[ QS = PS = PQ = 6 \text{ cm} \]

**Hint:** In an equilateral triangle, all sides are equal, which simplifies finding the length of the diagonal.

### Conclusion
The length of diagonal QS is 6 cm.

**Final Answer:** QS = 6 cm