The image contains a geometric problem involving a figure composed of trapezoids and rectangles. The goal is to calculate the heights of the trapezoids and the areas of the trapezoids and rectangles labeled A, B, C, D, and E, followed by the total area of the figure.

Let's break down the information:

• The dimensions for various segments are given:
  • The length of the entire bottom part is 21 cm (10 cm + 11 cm).
  • The leftmost vertical height is 11 cm.
  • Specific heights and lengths for parts of trapezoids and rectangles are also indicated.

We need to:

1. Find the heights of the trapezoids.
2. Compute the areas of the trapezoids and rectangles using their respective area formulas:
   • Area of a trapezoid: $A = \frac{1}{2} \times (b_1 + b_2) \times h$
   • Area of a rectangle: $A = \text{length} \times \text{width}$

I will now proceed to calculate the areas and heights step by step.

Height and Area Calculations:

1. Height of Trapezoid A:

   • $h_A = 11 \, \text{cm} - 7 \, \text{cm} = 4 \, \text{cm}$

2. Area of Trapezoid A:

   • $b_1 = 5 \, \text{cm}, b_2 = 5 \, \text{cm}, h_A = 4 \, \text{cm}$
   • $A_A = \frac{1}{2} \times (5 + 5) \times 4 = 20 \, \text{cm}^2$

3. Area of Rectangle B:

   • Width = 5 cm, Height = 7 cm.
   • $A_B = 5 \times 7 = 35 \, \text{cm}^2$

4. Height of Trapezoid C:

   • The height is directly given as 5 cm.

5. Area of Trapezoid C:

   • $b_1 = 7 \, \text{cm}, b_2 = 2 \, \text{cm}, h_C = 5 \, \text{cm}$
   • $A_C = \frac{1}{2} \times (7 + 2) \times 5 = 22.5 \, \text{cm}^2$

6. Area of Rectangle D:

   • Width = 11 cm, Height = 2 cm.
   • $A_D = 11 \times 2 = 22 \, \text{cm}^2$

7. Area of Rectangle E:

   • Width = 2 cm, Height = 9 cm.
   • $A_E = 2 \times 9 = 18 \, \text{cm}^2$

8. Total Area of the Figure:

   • Total area = $A_A + A_B + A_C + A_D + A_E$
   • Total = $20 + 35 + 22.5 + 22 + 18 = 117.5 \, \text{cm}^2$

Summary of Results:

• Height of Trapezoid A = 4 cm
• Area of Trapezoid A = 20 cm²
• Area of Rectangle B = 35 cm²
• Height of Trapezoid C = 5 cm
• Area of Trapezoid C = 22.5 cm²
• Area of Rectangle D = 22 cm²
• Area of Rectangle E = 18 cm²
• Total Area = 117.5 cm²

Would you like further details or clarifications on any step?

Here are 5 related questions to expand on this:

1. How do you calculate the area of composite figures?
2. What is the difference between calculating the area of a trapezoid and a rectangle?
3. How would the area change if one of the dimensions were increased by 2 cm?
4. What is the relationship between the perimeter and the area of this figure?
5. Can you calculate the height of a trapezoid given only its area and bases?

Tip: Always double-check units and dimensions when calculating area to avoid mistakes!