Introduction
In trapezium ABCD, where AB is parallel to DC, it is crucial to understand the properties of the diagonals and their intersection point O. The goal is to demonstrate that the areas of triangles AO, CO, BO, and DO are equal, meaning AO = CO = BO = DO.
Properties of Trapezium
- Parallel Sides: Since AB || DC, the angles formed by the diagonals with the bases are equal.
- Transversal: The diagonals AC and BD act as transversals, creating alternate interior angles that are equal.
Triangles in Trapezium
- Area Relationship: The area of a triangle is determined by 1/2 * base * height. In trapezium ABCD, the bases for triangles AO and CO are AB and DC, respectively.
- Heights: The height from point O to line AB is the same as from O to line DC, due to the parallel nature of the sides. Thus, the heights for triangles AO and CO are equal.
Equal Areas of Triangles
- Triangles AO and CO: Since both have the same base (AB = DC) and equal heights, their areas are equal.
- Triangles BO and DO: By similar reasoning, triangles BO and DO also have equal areas since they share the same height from O to the parallel lines.
Conclusion
- Final Assertion: Therefore, AO = CO = BO = DO, confirming that the areas of triangles formed by the diagonals in trapezium ABCD are equal. This property is essential in many geometric proofs and applications.
This understanding of trapeziums can significantly aid in solving complex geometrical problems and is a valuable concept for UPSC preparations.