The problem asks us to find the area of the trapezium given the shape in the image. Let's break this down step by step:

Given Information:

• The shape consists of a triangle on top of a trapezium.
• Area of the triangle: $24 \, \text{cm}^2$
• Height of the entire shape: $6 \, \text{cm}$
• Height of the trapezium: $2 \, \text{cm}$
• Base of the entire shape (bottom of the trapezium): $5 \, \text{cm}$

Step 1: Determine the height of the triangle

The total height of the shape is 6 cm, and the height of the trapezium is 2 cm. Therefore, the height of the triangle is: $\text{Height of the triangle} = 6 \, \text{cm} - 2 \, \text{cm} = 4 \, \text{cm}$

Step 2: Use the triangle’s area to find its base

The area of a triangle is given by the formula: $\text{Area of the triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ We know the area is 24 cm² and the height is 4 cm. Let $b_1$ be the base of the triangle: $24 = \frac{1}{2} \times b_1 \times 4$ Solving for $b_1$: $24 = 2 \times b_1 \quad \Rightarrow \quad b_1 = \frac{24}{2} = 12 \, \text{cm}$ Thus, the base of the triangle (and the top base of the trapezium) is 12 cm.

Step 3: Find the area of the trapezium

The formula for the area of a trapezium is: $\text{Area of the trapezium} = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}$ The two parallel sides are:

• The top base of the trapezium: $b_1 = 12 \, \text{cm}$
• The bottom base of the trapezium: $b_2 = 5 \, \text{cm}$ And the height of the trapezium is 2 cm.

Substituting these values into the formula: $\text{Area of the trapezium} = \frac{1}{2} \times (12 + 5) \times 2 = \frac{1}{2} \times 17 \times 2 = 17 \, \text{cm}^2$

Final Answer:

The area of the trapezium is $17 \, \text{cm}^2$.

Would you like further details on any step? Here are some related questions you might find interesting:

1. How do you calculate the area of a trapezium if the height is unknown?
2. Can the area formula for a trapezium be derived geometrically?
3. What are other methods for determining the area of composite shapes?
4. How do we solve for missing lengths if only the area is provided?
5. How does the concept of parallel lines impact the shape and area of a trapezium?

Tip: When working with composite shapes, always identify known and unknown values separately for each component.