ABCD is a trapezium in which AB parallel DC and its diagonals intersect each other at point O . show that AO/BO = CO/DO
A trapezoid (trapezium) is a quadrilateral in which one pair of opposite sides are parallel.
Answer: For a trapezium ABCD in which AB is parallel to DC and its diagonals intersect each other at point O we see that AO/BO = CO/DO
Let's prove AO/BO = CO/DO
Let's draw the diagram of trapezium ABCD as shown below:
Here, the sides DC and AB are parallel sides.
Now, consider the two triangles AOB and COD
Since DC is parallel to AB,
∠ODC = ∠OBA (Alternate interior angles)
Similarly, ∠OCD = ∠OAB (Alternate interior angles)
Also, ∠COD = ∠AOB (Vertically opposite angles).
Now according to AAA (angle-angle-angle) similarity criteria, the two triangles AOB and COD are similar.
Therefore, AO/CO = BO/DO
Now, interchange the positions of CO and BO in the above equation.
Hence, AO/BO = CO/DO