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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.

Determine the area of each composite figure.

Learn how to calculate the area of composite figures involving rectangles, triangles, and trapezoids with step-by-step solutions. Understand key geometry concepts and apply area formulas. Suitable for students in Grades 6-8.

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