Trapezium Area Calculation: Step-by-Step

The problem requires calculating the area of a trapezium given the lengths of its parallel sides and non-parallel sides.

Given values:

• Length of parallel side 1 ($a$): 10 cm
• Length of parallel side 2 ($b$): 20 cm
• Length of non-parallel sides: 10 cm (It's an isosceles trapezium)

Finding the Height of the Trapezium

To find the area, we first need the height ($h$) of the trapezium. We can find this by visualizing the isosceles trapezium. Draw perpendiculars from the endpoints of the shorter parallel side (10 cm) to the longer parallel side (20 cm).

This divides the longer base into three segments. The middle segment is equal to the shorter base (10 cm). The remaining length ($20 \text{ cm} - 10 \text{ cm} = 10 \text{ cm}$) is split equally between the two outer segments at the ends.

Length of each outer segment = $ \frac{b - a}{2} = \frac{20 \text{ cm} - 10 \text{ cm}}{2} = \frac{10 \text{ cm}}{2} = 5 \text{ cm} $.

Now, consider one of the right-angled triangles formed by a non-parallel side (hypotenuse = 10 cm), the height ($h$), and one of the outer segments (base = 5 cm).

Using the Pythagorean theorem ($ \text{hypotenuse}^2 = \text{height}^2 + \text{base}^2 $):

$ 10^2 = h^2 + 5^2 $

$ 100 = h^2 + 25 $

$ h^2 = 100 - 25 $

$ h^2 = 75 $

$ h = \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} \text{ cm} $

Calculating the Area

The formula for the area of a trapezium is:

$ \text{Area} = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height} $

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

Substitute the known values:

$ \text{Area} = \frac{1}{2} \times (10 \text{ cm} + 20 \text{ cm}) \times 5\sqrt{3} \text{ cm} $

$ \text{Area} = \frac{1}{2} \times (30 \text{ cm}) \times 5\sqrt{3} \text{ cm} $

$ \text{Area} = 15 \text{ cm} \times 5\sqrt{3} \text{ cm} $

$ \text{Area} = 75\sqrt{3} \text{ cm}^2 $

The area of the trapezium is $ 75\sqrt{3} \text{ cm}^2 $.