ABCD is a trapezium in which AB is parallel to CD, AB = 12 cm and CD = 24 cm. If O is the point of intersection of AC and BD, find the ratio of the area of triangle AOD to that of triangle COD.

Let the height of the trapezium be h. Triangles AOB and COD are similar. AB/CD = 1/2. So, height of triangle AOB / height of triangle COD = 1/2
So, height of triangle AOB = h/3 and height of triangle COD = 2h/3.

Area of triangle ADC = 1/2 * 24 * h
Area of triangle COD = 1/2 * 24 * 2h/3
Area of triangle AOD = Area of triangle ADC - Area of triangle COD = 1/2 * 24 * (h - 2h/3) = 1/2 * 24 * h/3

So, required ratio = 1/2 * 24 * h/3 / 1/2 * 24 * 2h/3 = 1/2