What is the area of the largest triangle that can be inscribed in a semicircle of radius r unit.

Options

A. `r^(2)` sq units
B. `(1)/(2)r^(2)` sq units
C. `2r^(2)` sq units
D. `sqrt(2)r^(2)` sq units

Explanation

To find the area of the largest triangle that can be inscribed in a semicircle of radius \( r \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- We have a semicircle with a radius \( r \). The diameter of the semicircle will be \( 2r \).

2. **Identifying the Triangle**:
- The largest triangle that can be inscribed in the semicircle will have its base along the diameter of the semicircle. This means the base of the triangle is equal to the diameter, which is \( 2r \).

3. **Determining the Height**:
- The height of the triangle will be the distance from the apex of the triangle (the point on the semicircle) to the base (the diameter). The maximum height occurs when the apex is at the highest point of the semicircle, which is equal to the radius \( r \).

4. **Calculating the Area**:
- The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
- Substituting the values we have:
\[ A = \frac{1}{2} \times (2r) \times r \]
- Simplifying this gives:
\[ A = \frac{1}{2} \times 2r^2 = r^2 \]

5. **Conclusion**:
- Therefore, the area of the largest triangle that can be inscribed in a semicircle of radius \( r \) is \( r^2 \) square units.

### Final Answer:
The area of the largest triangle that can be inscribed in a semicircle of radius \( r \) is \( r^2 \) square units.

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