2026 DSS P2 Q9

In the diagram below, \(ABCD\) is a trapezium such that \(AD\) is parallel to \(BC\). Lines \(AC\) and \(BD\) intersect at point \(X\).

1. Show that triangle \(CBX\) is similar to triangle \(ADX\).[2]
2. The ratio \(AD:BC=2:5\).
   Find the ratio of
   1. area of triangle \(ADX\) : area of triangle \(CBX\),[1]
   2. area of triangle \(ABD\) : area of triangle \(ABC\),[1]
   3. area of triangle \(ABX\) : area of trapezium \(ABCD\).[2]
3. Given that the area of triangle \(ABX\) is \(8\text{ cm}^2\), find the area of the shaded region.[2]
4. State an additional condition for triangle \(BAX\) and triangle \(CDX\) to be congruent.
   Support your answer with reasons.[2]