The angle subtended by side of a parallelogram with pair of other parallel lines is `150^(@)`. If distance between parallel sides PQ and SR is 20 cm then what is the length of side RQ ?

Options

A. 40 cm
B. 50 cm
C. 60 cm
D. 70 cm

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript and apply the necessary mathematical principles.

### Step-by-Step Solution:

1. **Understanding the Parallelogram**:
- We have a parallelogram \( PQRS \) where \( PQ \) is parallel to \( SR \) and \( PS \) is parallel to \( QR \).
- The angle subtended by the side \( PQ \) with the parallel lines \( SR \) is \( 150^\circ \).

2. **Finding the Related Angles**:
- The angle \( \angle SPQ \) can be found using the property that the sum of angles on a straight line is \( 180^\circ \).
- Therefore, \( \angle SPQ = 180^\circ - 150^\circ = 30^\circ \).

3. **Drawing the Altitude**:
- Draw a perpendicular from point \( S \) to line \( PQ \), meeting it at point \( M \).
- The distance between the parallel sides \( PQ \) and \( SR \) is given as \( 20 \, \text{cm} \). Thus, \( SM = 20 \, \text{cm} \).

4. **Using Trigonometric Ratios**:
- In triangle \( SPM \), we can use the tangent function:
\[ \tan(30^\circ) = \frac{SM}{PM} \]
- We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \).
- Substituting the known values:
\[ \frac{1}{\sqrt{3}} = \frac{20}{PM} \]
- Rearranging gives:
\[ PM = 20 \sqrt{3} \, \text{cm} \]

5. **Applying Pythagorean Theorem**:
- Now, we can find the length of \( SP \) using the Pythagorean theorem:
\[ SP = \sqrt{SM^2 + PM^2} \]
- Substituting the values:
\[ SP = \sqrt{20^2 + (20\sqrt{3})^2} \]
- Calculating:
\[ SP = \sqrt{400 + 1200} = \sqrt{1600} = 40 \, \text{cm} \]

6. **Finding the Length of Side \( RQ \)**:
- Since \( PQRS \) is a parallelogram, opposite sides are equal. Therefore, the length of side \( RQ \) is equal to the length of side \( SP \).
- Thus, \( RQ = SP = 40 \, \text{cm} \).

### Final Answer:
The length of side \( RQ \) is \( 40 \, \text{cm} \).