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!