Diagonals Bisect Each Other (Motivate) MCQs Quiz | Class 9

This quiz covers Class IX Mathematics (Code 041), Unit IV: Geometry, specifically focusing on the property and converse statements that diagonals of a parallelogram bisect each other. Test your understanding of quadrilateral properties, submit your answers, and download the PDF solution sheet for offline revision.

Understanding Diagonals in Quadrilaterals

In Class 9 Geometry, a fundamental property of parallelograms is that their diagonals bisect each other. This implies that the point of intersection of the two diagonals divides each diagonal into two equal parts. This property is crucial for proving that a given quadrilateral is a parallelogram.

Key Properties

  • Property: The diagonals of a parallelogram bisect each other.
  • Converse: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
  • Rhombus: Diagonals bisect each other at right angles (90 degrees).
  • Rectangle: Diagonals are equal in length and bisect each other.
  • Square: Diagonals are equal, bisect each other, and intersect at right angles.

Comparison Table

Quadrilateral Diagonals Bisect? Diagonals Equal? Intersect at 90 degrees?
Parallelogram Yes No No
Rectangle Yes Yes No
Rhombus Yes No Yes
Square Yes Yes Yes

Proof Concept (Motivation)

To motivate the proof that diagonals bisect each other, we typically compare the two triangles formed by the intersection of diagonals and opposite sides. By using the Angle-Side-Angle (ASA) congruence criterion, we can show that the segments of the diagonals are equal corresponding parts of congruent triangles.

Quick Revision List

  1. Intersection point is the midpoint of both diagonals.
  2. This property helps find unknown coordinates in coordinate geometry.
  3. It distinguishes parallelograms from trapeziums (where diagonals generally do not bisect).
  4. In a kite, only one diagonal is bisected by the other.

Extra Practice Questions

  • 1. If one diagonal of a quadrilateral is the perpendicular bisector of the other, is it necessarily a rhombus? (Answer: No, it could be a kite).
  • 2. In parallelogram ABCD, if diagonals intersect at O and AC = 12 cm, find AO. (Answer: 6 cm).
  • 3. Prove that a parallelogram with equal diagonals is a rectangle.
  • 4. If diagonals of a quadrilateral bisect each other at right angles, what specific type is it? (Answer: Rhombus or Square).
  • 5. Can a trapezium have diagonals that bisect each other? (Answer: No, never).