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This question asks to show that a quadrilateral \( ABCD \) is a trapezium, given the condition:

\[
\frac{AO}{BO} = \frac{CO}{DO}
\]

where the diagonals \( AC \) and \( BD \) intersect at point \( O \).

### Step-by-Step Solution:

We need to prove that \( ABCD \) is a **trapezium**, which means one pair of opposite sides must be parallel (in this case, either \( AB \parallel DC \) or \( AD \parallel BC \)).

### Step 1: Use the Converse of the Basic Proportionality Theorem (BPT)

The condition \( \frac{AO}{BO} = \frac{CO}{DO} \) implies that the diagonals divide each other proportionally at point \( O \).

The **Converse of the Basic Proportionality Theorem** states:
>If a line divides two sides of a triangle in the same ratio, then that line is parallel to the third side.

Here, let’s apply this to two triangles:
- Consider \( \triangle ABD \).
- The condition \( \frac{AO}{BO} = \frac{CO}{DO} \) tells us that the line passing through \( O \), which is the diagonal \( AC \), divides the sides \( AB \) and \( BD \) proportionally.

By the **Converse of BPT**, this means that \( AB \parallel DC \).

### Step 2: Conclude That \( ABCD \) is a Trapezium

Since \( AB \parallel DC \), it follows that the quadrilateral \( ABCD \) is a **trapezium** by definition (one pair of opposite sides, \( AB \) and \( DC \), are parallel).

### Conclusion:

We have used the **Converse of the Basic Proportionality Theorem** to show that \( AB \parallel DC \). Therefore, \( ABCD \) is a trapezium.

Would you like any clarification on this method? Here are some questions to explore further:

1. Can you apply the Basic Proportionality Theorem to any other quadrilateral?
2. How would you prove that a different pair of sides is parallel in a similar problem?
3. What happens if \( \frac{AO}{BO} \neq \frac{CO}{DO} \)? Would it still be a trapezium?
4. How does the converse of BPT differ from the direct BPT application?
5. Can you use this theorem in other geometric shapes like parallelograms or rectangles?

### Tip:
Whenever you see that diagonals or lines divide sides of triangles or quadrilaterals in the same ratio, think about using **BPT or its converse** to check for parallel lines!

The diagonals of a quadrilateral ABCD intersect each other at the point O such that AO/BO = CO/DO. Show that ABCD is a trapezium.

Learn how to prove that a quadrilateral ABCD is a trapezium, using the Converse of the Basic Proportionality Theorem when the diagonals intersect at point O and divide proportionally. Suitable for students in Grades 9-10.

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