Question

If the diagonals of a quadrilateral divide each other proportionally prove that it is trapezium

Answer 1

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Answer 2

The diagonals of the quadrilateral divide each other proportionally it proves that it forms the trapezium.

Solution:

As the figure shown above the diagonals are BD and AC for the quadrilateral ABCD.

Hence $\bold{\frac{A O}{O C}=\frac{D O}{O B}}$

In a triangle, AOD and COD the angles of AOD and COD are adjacent to each other.

As the diagonals divides the quadrilateral proportionally the sum angles of AOD and COD $\ =180^{\circ}$ .

In triangle AOB and BOC the angles of AOB and BOC are adjacent to each other.

As the diagonals divides the quadrilateral proportionally the sum angles of AOD and COD $\ =180^{\circ}$ .

Due to the proportionality of the diagonals the angle OBC and ADO are equal and so as the angle of DAO and OCB.

Other than the above speculations, the line AD is parallel to CB.