State ‘true’ or ‘false’:

The diagonals of a parallelogram bisect each other at right angle:

• a) True
• b) False

The statement is false. In a parallelogram, the diagonals do indeed bisect each other, but they do not necessarily intersect at right angles.

Understanding Parallelograms and Their Properties

To grasp why this is the case, let’s delve into the properties of parallelograms. A parallelogram is defined as a four-sided figure (quadrilateral) where opposite sides are both equal in length and parallel. This unique structure leads to several interesting properties, particularly concerning its diagonals.

Diagonal Bisection

One of the key characteristics of a parallelogram is that its diagonals bisect each other. This means that if you take the two diagonals of a parallelogram, they will intersect at a point that divides each diagonal into two equal segments. For example, if you label the vertices of a parallelogram as A, B, C, and D, the diagonals AC and BD will intersect at point E, making AE = EC and BE = ED.

Angle of Intersection

However, the angle at which these diagonals intersect is not fixed at 90 degrees. In fact, the diagonals of a rectangle (a specific type of parallelogram) do intersect at right angles, but this is not true for all parallelograms. For instance, in a rhombus (another type of parallelogram), the diagonals do intersect at right angles, but in a general parallelogram, they can intersect at any angle.

Visualizing with Examples

• Rectangle: The diagonals bisect each other and meet at right angles.
• Rhombus: The diagonals bisect each other and also meet at right angles.
• General Parallelogram: The diagonals bisect each other but do not necessarily meet at right angles.

To visualize this, imagine a rectangle drawn on a piece of paper. If you draw the diagonals, they will cross each other at the center, forming four right angles. Now, take a parallelogram that is not a rectangle, like a slanted shape. The diagonals will still cross at the center, but the angles formed will not be right angles.

Conclusion

In summary, while the diagonals of a parallelogram do bisect each other, they do not do so at right angles unless the parallelogram has specific properties, like being a rectangle or a rhombus. This distinction is crucial for understanding the broader category of quadrilaterals and their unique characteristics.