In the trapezium  ABCD, ∠BAE  =  30∘, ∠CDF  =  45∘, BC  =  6cm  and  AB  =  12cm . Find the area of trapezium:

To find the area of the trapezium ABCD, we need to find the height of the trapezium. We can do this by using the given angles and the lengths of the sides.

Step 1: Find the height of the trapezium The height of the trapezium can be found by using the properties of the right triangle. In triangle ABE, angle BAE is 30 degrees and AB is the hypotenuse which is 12 cm. So, the height (BE) can be found by using the sine of the angle:

sin(30) = BE / AB BE = AB * sin(30) BE = 12cm * 0.5 = 6cm

Step 2: Find the length of the base (CD) In triangle CDF, angle CDF is 45 degrees and CF is the height of the trapezium which is 6 cm. So, the length of the base (CD) can be found by using the tangent of the angle:

tan(45) = CF / CD CD = CF / tan(45) CD = 6cm / 1 = 6cm

Step 3: Find the area of the trapezium The area of a trapezium is given by the formula:

Area = 1/2 * (sum of parallel sides) * height Area = 1/2 * (AB + CD) * BE Area = 1/2 * (12cm + 6cm) * 6cm Area = 1/2 * 18cm * 6cm = 54 cm²

So, the area of the trapezium ABCD is 54 cm².