In this problem, you are asked to determine the areas of two different composite figures labeled as (a) and (b). Let's go through the calculations for each figure step-by-step.

Figure (a):

This is a composite figure that consists of a rectangle, a triangle, and a trapezoid.

Step 1: Identify the shapes and their dimensions.

1. Rectangle: Dimensions are 4 cm by 8 cm.
2. Trapezoid: The height is 5 cm, the parallel sides are 10 cm and 4 cm.
3. Triangle: The height is 4 cm, and the base is 6 cm.

Step 2: Calculate the area of each shape.

1. Rectangle: $\text{Area of rectangle} = \text{length} \times \text{width} = 8 \, \text{cm} \times 4 \, \text{cm} = 32 \, \text{cm}^2$

2. Trapezoid: The area of a trapezoid is given by: $\text{Area of trapezoid} = \frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height}$ Substituting the values: $\text{Area of trapezoid} = \frac{1}{2} \times (10 \, \text{cm} + 4 \, \text{cm}) \times 5 \, \text{cm} = \frac{1}{2} \times 14 \, \text{cm} \times 5 \, \text{cm} = 35 \, \text{cm}^2$

3. Triangle: The area of a triangle is given by: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ Substituting the values: $\text{Area of triangle} = \frac{1}{2} \times 6 \, \text{cm} \times 4 \, \text{cm} = \frac{1}{2} \times 24 \, \text{cm}^2 = 12 \, \text{cm}^2$

Step 3: Add the areas to get the total area.

$\text{Total area of figure (a)} = 32 \, \text{cm}^2 + 35 \, \text{cm}^2 + 12 \, \text{cm}^2 = 79 \, \text{cm}^2$

Figure (b):

This composite figure consists of three trapezoids.

Step 1: Identify the trapezoids and their dimensions.

1. Top trapezoid: Parallel sides are 4 cm and 6 cm, height is 3 cm.
2. Middle trapezoid: Parallel sides are 6 cm and 8 cm, height is 3 cm.
3. Bottom trapezoid: Parallel sides are 4 cm and 8 cm, height is 3 cm.

Step 2: Calculate the area of each trapezoid.

1. Top trapezoid: $\text{Area} = \frac{1}{2} \times (4 \, \text{cm} + 6 \, \text{cm}) \times 3 \, \text{cm} = \frac{1}{2} \times 10 \, \text{cm} \times 3 \, \text{cm} = 15 \, \text{cm}^2$

2. Middle trapezoid: $\text{Area} = \frac{1}{2} \times (6 \, \text{cm} + 8 \, \text{cm}) \times 3 \, \text{cm} = \frac{1}{2} \times 14 \, \text{cm} \times 3 \, \text{cm} = 21 \, \text{cm}^2$

3. Bottom trapezoid: $\text{Area} = \frac{1}{2} \times (4 \, \text{cm} + 8 \, \text{cm}) \times 3 \, \text{cm} = \frac{1}{2} \times 12 \, \text{cm} \times 3 \, \text{cm} = 18 \, \text{cm}^2$

Step 3: Add the areas to get the total area.

$\text{Total area of figure (b)} = 15 \, \text{cm}^2 + 21 \, \text{cm}^2 + 18 \, \text{cm}^2 = 54 \, \text{cm}^2$

Summary:

• The total area of figure (a) is 79 cm².
• The total area of figure (b) is 54 cm².

Would you like further clarification on any steps, or additional details on how these areas were calculated?

Here are 5 related questions to further your understanding:

1. How would the total area change if the base of the triangle in figure (a) was increased by 2 cm?
2. What is the formula for the area of a trapezoid, and why is it useful in composite figures?
3. How can you decompose irregular shapes to find their area more easily?
4. What happens to the area of a figure if both the length and width are doubled?
5. How would you calculate the perimeter of figure (b)?

Tip: When working with composite shapes, breaking the figure into familiar shapes (like triangles, rectangles, trapezoids) simplifies area calculations.