Area Of A Triangle By Heron S Formula

To solve the problem, we need to find the ratio of areas of triangles COD and AOB in a trapezium where AB || DC and AB = 2CD.

1. Understanding the Geometry:

In trapezium ABCD with AB || DC and diagonals AC and BD intersecting at O, triangles AOB and COD are formed by the intersection of diagonals.

2. Using Similar Triangles and Area Concept:

Since AB || DC and the diagonals intersect at O, triangles AOB and COD are between the same set of diagonals and share a common height from point O (as diagonals intersect and form vertically opposite angles).

3. Area of Triangle Using Base and Height:

The area of a triangle is given by:

For both triangles AOB and COD, the height from O to bases AB and CD respectively is the same (as both share the same set of diagonals).

4. Ratio of Areas:

Since height is common, the ratio of areas is simply the ratio of the bases AB and CD:

Therefore, the ratio of areas of COD to AOB is:

5. Final Ratio:

We are asked the ratio of COD : AOB, hence:

Final Answer:

The ratio of the areas of triangles COD and AOB is .