Trapezium ABCD, AB || CD, Diagonals intersect at point O. How to find area of Triangle BOC 

In the trapezium ABCD, where \( AB \parallel CD \) and the diagonals intersect at point \( O \), the area of triangle \( BOC \) can be calculated using the following approach:

### Step-by-Step Solution:

1. **Identify Key Properties:**
   - \( AB \parallel CD \), which implies the trapezium is symmetric in some sense.
   - The diagonals of a trapezium divide it into smaller triangles that share certain area ratios.

2. **Diagonal Division Property:**
   The diagonals of a trapezium divide each other proportionally in the ratio of the lengths of the parallel sides. Therefore, we can say:
   \[
   \frac{AO}{OC} = \frac{AB}{CD}
   \]
   This means the diagonals are divided in the ratio of the lengths of the parallel sides \( AB \) and \( CD \).

3. **Area of Triangles:**
   The triangles \( AOD \), \( BOC \), \( AOB \), and \( COD \) are formed by the diagonals. Since \( AB \parallel CD \), triangles \( BOC \) and \( AOD \) are similar, and the area ratio of triangles \( AOB \) and \( COD \) will also be proportional to the sides.

4. **Formula for Area of \( \Delta BOC \):**
   The total area of the trapezium is the sum of the areas of these four triangles. Since \( O \) divides the diagonals proportionally, the area of triangle \( BOC \) can be found using the proportionality:
   \[
   \text{Area of } \Delta BOC = \frac{AB}{AB + CD} \times \text{Area of the trapezium}
   \]

Thus, the area of \( \Delta BOC \) depends on the lengths of the parallel sides \( AB \) and \( CD \), and the area of the entire trapezium.

### Clarifications Needed:
- The exact dimensions of \( AB \), \( CD \), and the height or the area of the trapezium are required to compute the specific value of the area of triangle \( BOC \).

Would you like to provide these values or have any other questions?

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### Further Exploration Questions:
1. What is the ratio of areas of triangles \( AOB \) and \( COD \)?
2. How do the diagonals divide each other in trapezium ABCD?
3. What is the total area of the trapezium, if the lengths of \( AB \), \( CD \), and height are known?
4. How to compute the area of \( \Delta BOC \) if only the lengths of the diagonals are known?
5. What is the significance of the ratio \( \frac{AO}{OC} \) in solving other parts of the trapezium?

### Tip:
In a trapezium, using the ratio of the lengths of parallel sides often simplifies the problem involving diagonals.

Trapezium ABCD, AB || CD, Diagonals intersect at point O. How to find area of Triangle BOC?

Learn how to calculate the area of triangle BOC in a trapezium where AB is parallel to CD, and diagonals intersect at point O. The solution involves the proportional division of diagonals and applying geometric area formulas. Suitable for students in Grades 9-12.

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