ABCD is a trapezium - OCR - GCSE Maths - Question 20 - 2020 - Paper 1

The area of a trapezium is calculated using the formula:

$\text{Area} = \frac{1}{2} \times (a + b) \times h$

where a and b are the lengths of the parallel sides, and h is the height.

Here, a = AB = 24 cm and b = DC = 12 cm.

To find the height (h), we will need to use the properties of the trapezium. From the dimensions we have (AD = 10 cm, which acts as the leg of the trapezoid), we can use the Pythagorean theorem if we find the height between the two parallel sides (assuming we dropped perpendiculars from points D and C to line AB).

Using some geometry or trigonometry, we establish that the height will be under the assumption of a right angle triangle formed by dropping perpendiculars. If we assume one leg to be h and the adjacent is part of the triangle:

$h = \sqrt{AD^2 - \left(\frac{|AB - DC|}{2}\right)^2}$

Substituting values gives:

1. Compute the length difference:
   • $\frac{|24 - 12|}{2} = 6$
2. Now compute h:
   • $h = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8$

Now substitute into the area formula:

$\text{Area} = \frac{1}{2} \times (24 + 12) \times 8 = \frac{1}{2} \times 36 \times 8 = 144 \text{ cm}^2$