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The problem involves a geometric configuration with a semicircle and a triangle inscribed within it. Here's a summary of the key details from the image:

- There is a semicircle with center \( O \), and a radius of \( \frac{2}{10} \) units (which simplifies to \( 0.2 \)).
- The points \( A \), \( B \), and \( C \) are described:
  - \( A \) and \( C \) are on the circumference of the circle.
  - \( B \) is within the semicircle.
- The triangle \( ABC \) is a right triangle with \( \angle ABC = 90^\circ \).
- We are provided:
  - \( AB = 8 \)
  - \( BC = 4 \)
  - The segment \( OB = 1 \).

The problem asks to find the area of the sector defined by angle \( \frac{3\pi}{2} \).

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### Solution Outline:

1. **Interpret the geometry**: We need to interpret the given triangle, semicircle, and radius conditions carefully, focusing on triangle properties and the angle formed by the sector.
2. **Determine the area** of the sector defined by the angle \( \frac{3\pi}{2} \) (which is \( 270^\circ \)), using the known radius of the semicircle and basic sector area formula.
   
Let me calculate this for you.The area of the sector defined by the angle \( \frac{3\pi}{2} \) (270°) is approximately \( 2.356 \, \text{square units} \).

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1. How is the formula for the area of a sector derived?
2

The problem involves a semicircle with center O, radius \( \frac{2}{10} \), and triangle ABC, where AB = 8, BC = 4, OB = 1, and \( \angle ABC = 90^{\circ} \). Find the area of the sector defined by the angle \( \frac{3\pi}{2} \).

Solve a geometry problem involving a semicircle with a radius of \( 0.2 \), a right triangle inscribed in the circle, and calculate the area of a sector defined by an angle of \( \frac{3\pi}{2} \). Suitable for high school students.

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