Here is trapezium ABCD - Edexcel - GCSE Maths - Question 19 - 2021 - Paper 1

The area of a trapezium is given by the formula:\n$\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}$\nPlugging in the known values, we have:\n$66 = \frac{1}{2} \times (2x + 3x) \times h$\nwhere ( h ) represents the height.

From triangle BCD, we can use the sine function to find the height. Considering angle 30°:\n$\sin(30°) = \frac{h}{6}$\nThis results in:\nh = 6 \times \sin(30°) = 6 \times \frac{1}{2} = 3 \text{ cm}.

Now substituting h back into the area equation, we have:\n$66 = \frac{1}{2} \times (5x) \times 3$\nSimplifying gives us:\n$66 = \frac{15x}{2}$\nThus, multiplying both sides by 2 results in:\n( 132 = 15x ) or ( x = \frac{132}{15} = 8.8 ).

The length of AB can be found by substituting the value of x back into the equation for AB:\n( AB = 2x = 2 \times 8.8 = 17.6 \text{ cm} ).