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

Options

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

Answer

To find the length of QS in the rhombus PQRS where the side length PQ = 6 cm and the angle PQR = 120°, we can follow these steps:

### Step 1: Understand the properties of the rhombus
In a rhombus, all sides are equal in length. Therefore, we have:
- PQ = QR = RS = SP = 6 cm

### Step 2: Analyze the given angle
We are given that angle PQR = 120°. Since RQ is parallel to SP, we can use the properties of angles formed by a transversal.

### Step 3: Find the angle PSQ
Since RQ is parallel to SP and PQ is a transversal, we can find angle PSQ. The sum of the interior angles on the same side of the transversal is 180°:
- Angle PQR + Angle PSQ = 180°
- 120° + Angle PSQ = 180°
- Angle PSQ = 180° - 120° = 60°

### Step 4: Analyze triangle PQS
Now, we look at triangle PQS. We know:
- Angle PQS = Angle PSQ = 60° (from the previous step)
- Angle PQR = 120° (given)

### Step 5: Find the remaining angle in triangle PQS
The sum of angles in a triangle is 180°. Therefore, we can find the third angle, angle QPS:
- Angle PQS + Angle PSQ + Angle QPS = 180°
- 60° + 60° + Angle QPS = 180°
- Angle QPS = 180° - 120° = 60°

### Step 6: Conclude the type of triangle
Since all angles in triangle PQS are 60°, triangle PQS is an equilateral triangle.

### Step 7: Find the length of QS
In an equilateral triangle, all sides are equal. Therefore:
- QS = PQ = 6 cm

Thus, the length of QS is **6 cm**.

### Final Answer
The length of QS is **6 cm**.

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