Parallelogram: Detailed Explanation
A parallelogram is a quadrilateral (four-sided figure) that is characterized by having opposite sides that are both parallel and equal in length. It is a special type of polygon with several important geometric properties.
Properties of a Parallelogram:
- Opposite sides are equal and parallel: A parallelogram has two pairs of opposite sides, and each pair is equal in length and parallel.
- Opposite angles are equal: The angles opposite each other in a parallelogram are equal in measure.
- Adjacent angles are supplementary: Any two angles that are next to each other (adjacent angles) sum up to 180°.
- Diagonals bisect each other: The diagonals of a parallelogram cut each other in half, meaning they bisect each other.
- Area and Perimeter:
- Area = Base × Height (The base is the length of any side, and the height is the perpendicular distance from that side to its opposite side).
- Perimeter = 2 × (Base + Side length).
- Symmetry:
- A parallelogram does not generally have line symmetry (except in special cases like a rhombus or rectangle).
- It has rotational symmetry at 180°.
Comparison of Parallelogram with Other Figures:
| Figure | Similarities with Parallelogram | Differences from Parallelogram |
|---|---|---|
| Square | – Opposite sides are parallel and equal. – Diagonals bisect each other. | – In a square, all sides are equal, and all angles are 90°. – The diagonals of a square bisect at right angles and are equal in length. |
| Rectangle | – Opposite sides are parallel and equal. – Diagonals bisect each other. | – A rectangle has all angles equal to 90°, while a parallelogram does not necessarily have right angles. – Diagonals of a rectangle are equal, unlike a general parallelogram. |
| Rhombus | – Opposite sides are parallel and equal. – Diagonals bisect each other at 90°. | – In a rhombus, all sides are equal, unlike a general parallelogram. – Diagonals of a rhombus bisect at right angles, which is not a requirement in a general parallelogram. |
| Trapezium (Trapezoid) | – Both are quadrilaterals. | – A trapezium has only one pair of parallel sides, while a parallelogram has two pairs. – The sides of a trapezium are not necessarily equal, and the angles can vary significantly. |
Parallelogram vs Square:
- Commonality: Both have opposite sides that are parallel and equal. In both, the diagonals bisect each other.
- Difference: In a square, all sides are equal, and all angles are 90°. The diagonals of a square are also equal and bisect each other at 90°, whereas in a general parallelogram, the angles are not necessarily 90°, and the diagonals may not be equal or intersect at right angles.
Parallelogram vs Rectangle:
- Commonality: Both have opposite sides that are parallel and equal, and the diagonals bisect each other.
- Difference: A rectangle has all angles equal to 90°, whereas a parallelogram does not necessarily have right angles. Also, the diagonals of a rectangle are equal, but in a parallelogram, they are generally unequal.
Parallelogram vs Rhombus:
- Commonality: Both have opposite sides that are parallel, and the diagonals bisect each other. Both are types of parallelograms.
- Difference: In a rhombus, all sides are equal, but the angles are not necessarily 90°. The diagonals of a rhombus bisect at right angles, which is not always the case for a general parallelogram.
Parallelogram vs Trapezium:
- Commonality: Both are quadrilaterals (four-sided figures).
- Difference: In a parallelogram, opposite sides are parallel and equal, while in a trapezium, only one pair of sides is parallel. The angles and sides are also not necessarily equal in a trapezium, whereas a parallelogram has equal opposite sides and angles.
Special Types of Parallelograms:
- Rectangle: A parallelogram with all angles equal to 90°.
- Rhombus: A parallelogram with all sides equal in length.
- Square: A parallelogram with all sides equal and all angles equal to 90°.
Real-Life Examples of a Parallelogram:
- Laptop screens: Many laptop screens are shaped like rectangles, which are special types of parallelograms.
- Tabletops: Certain tables or desks are parallelogram-shaped.
- Bridges: Some structural designs in bridges use parallelogram shapes in their framework.
Important Formulas:
- Area = Base × Height.
- Perimeter = 2 × (Base + Side length).
- Diagonals: In a parallelogram, the diagonals bisect each other but are not necessarily equal or perpendicular.