quadrilateral ABCD is a trapezium iff its diagonals intersect each other in the same ratio .proof this

Question

quadrilateral ABCD is a trapezium iff its diagonals intersect each other in the same ratio .proof this

Answer 1

google is your friend. Try here:

https://www.math.washington.edu/~king/coursedir/m444a00/syl/class/trapezoids/Trapezoids.html

Answer 2

To prove that a quadrilateral ABCD is a trapezium if and only if its diagonals intersect each other in the same ratio, let's break it down into two separate parts and prove each direction individually.

Part 1: If ABCD is a trapezium, then its diagonals intersect each other in the same ratio.
Assume that ABCD is a trapezium with bases AB and CD. Let the diagonals AC and BD intersect at point O. We need to prove that the diagonals intersect in the same ratio.

In triangle AOB, apply the Intercept Theorem. According to the theorem, if a pair of parallel lines (AB and CD) is intersected by a transversal (AC or BD), then the segments intercepted are proportional. In this case, we have:

AO/OB = AD/BC

In triangle COD, using the same theorem:

DO/OC = AD/BC

From these two equations, we can conclude that:

AO/OB = DO/OC

Thus, the diagonals AC and BD intersect each other at point O in the same ratio, which proves one direction of the statement.

Part 2: If the diagonals of ABCD intersect each other in the same ratio, then ABCD is a trapezium.
Assume that the diagonals AC and BD of quadrilateral ABCD intersect at point O, and they intersect each other in the same ratio.

Since the diagonals intersect at point O, we can apply the Similar Triangles Theorem. According to the theorem, if two sets of parallel lines (AB || CD and AC || BD) are intersected by a transversal (AC or BD) such that the corresponding segments are proportional, then the sets of lines are parallel. In this case, we have:

AO/OB = DO/OC

From this equation, it implies that AB || CD.

Hence, if the diagonals AC and BD intersect each other in the same ratio, the quadrilateral ABCD is a trapezium.

By proving both directions, we have established that a quadrilateral ABCD is a trapezium if and only if its diagonals intersect each other in the same ratio.