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Let's go through each of the exercises to calculate the areas of the indicated colored regions:

### Exercise 19:
This is a composite shape made of a trapezoid on top of a semicircle.

- **Semicircle**:
  The radius is \( r = 8 \) units.
  The area of a full circle is \( A_{\text{circle}} = \pi r^2 = \pi (8^2) = 64\pi \).
  The area of the semicircle is half of that:
  \[
  A_{\text{semicircle}} = \frac{1}{2} \times 64\pi = 32\pi.
  \]

- **Trapezoid**:
  The formula for the area of a trapezoid is:
  \[
  A_{\text{trapezoid}} = \frac{1}{2} (b_1 + b_2) \times h.
  \]
  Here, the bases are \( b_1 = 8 \) units, \( b_2 = 10 \) units, and the height is \( h = 8 \) units. So, the area is:
  \[
  A_{\text{trapezoid}} = \frac{1}{2} (8 + 10) \times 8 = \frac{1}{2} \times 18 \times 8 = 72 \text{ square units}.
  \]

- **Total area** (colored region):
  The total colored area is the sum of the semicircle and the trapezoid:
  \[
  A_{\text{total}} = 32\pi + 72.
  \]

### Exercise 20:
This is a quarter circle with radius \( r = 5 \) cm.

- The area of a full circle is \( A = \pi r^2 = \pi (5^2) = 25\pi \) square cm.
- The area of a quarter circle is:
  \[
  A_{\text{quarter}} = \frac{1}{4} \times 25\pi = \frac{25\pi}{4} \text{ square cm}.
  \]

### Exercise 21:
This is a quarter circle of radius \( r = 12 \) cm, minus the area of a triangle formed inside.

- **Quarter circle**:
  The area of a full circle is \( A = \pi r^2 = \pi (12^2) = 144\pi \) square cm.
  The area of the quarter circle is:
  \[
  A_{\text{quarter}} = \frac{1}{4} \times 144\pi = 36\pi \text{ square cm}.
  \]

- **Triangle**:
  The triangle is a right triangle with both legs measuring \( 12 \) cm. The area is:
  \[
  A_{\text{triangle}} = \frac{1}{2} \times 12 \times 12 = 72 \text{ square cm}.
  \]

- **Colored area**:
  The shaded region is the difference between the quarter circle and the triangle:
  \[
  A_{\text{shaded}} = 36\pi - 72 \text{ square cm}.
  \]

### Exercise 22:
This is an annular (ring-shaped) region, where the outer circle has a radius of \( 1.7 \) cm and the inner circle has a radius of \( 0.3 \) cm.

- **Outer circle**:
  The area of the outer circle is:
  \[
  A_{\text{outer}} = \pi (1.7)^2 = 2.89\pi \text{ square cm}.
  \]

- **Inner circle**:
  The area of the inner circle is:
  \[
  A_{\text{inner}} = \pi (0.3)^2 = 0.09\pi \text{ square cm}.
  \]

- **Colored area** (annular region):
  The area of the ring is the difference between the outer and inner circles:
  \[
  A_{\text{annular}} = 2.89\pi - 0.09\pi = 2.8\pi \text{ square cm}.
  \]

### Exercise 23:
This involves an equilateral triangle with sides of \( 2 \) units, and some regions inside it are shaded.

- **Equilateral triangle area**:
  The area of an equilateral triangle is given by the formula:
  \[
  A_{\text{triangle}} = \frac{\sqrt{3}}{4} s^2,
  \]
  where \( s \) is the side length. Here, \( s = 2 \), so:
  \[
  A_{\text{triangle}} = \frac{\sqrt{3}}{4} \times 2^2 = \sqrt{3} \text{ square units}.
  \]

Since this exercise has specific shaded regions, I would need to calculate the individual areas of those sections. Could you clarify whether it's the shaded regions or the remaining regions you'd like calculated?

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Do you have any questions or would you like more detailed steps for any specific part?

Here are 5 related questions you might consider:
1. What is the formula for the area of a sector of a circle?
2. How do you calculate the area of composite shapes?
3. Can the Pythagorean theorem be used to solve for the sides of triangles in problems like these?
4. What is the significance of the radius in determining the area of a circle or semicircle?
5. How can symmetry be useful when finding areas of shapes with uniform sections?

**Tip**: Always remember to ensure that units are consistent when calculating areas, particularly when working with composite shapes.

Calculate the area of the colored regions in the given geometric shapes.

Learn how to calculate the areas of composite shapes, including semicircles, trapezoids, quarter circles, and equilateral triangles. Step-by-step solutions for finding areas of colored regions in shapes. Suitable for Grades 9-10 geometry students.

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