The area of the largest triangle that can be inscribed in a semicircle of radius r is :

Options

A. `r^(2) cm^(2) `
B. `((r)/(3))^(2) cm^(2) `
C. `r sqrt(2) cm^(2) `
D. `3 sqrt(3r) cm^(2) `

Answer

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

### Step 1: Understand the Geometry
We have a semicircle with a radius \( r \). The center of the semicircle is denoted as point \( O \). The triangle inscribed in the semicircle will have its base along the diameter of the semicircle and its apex at the circumference.

### Step 2: Identify the Triangle's Base and Height
The base of the triangle will be the diameter of the semicircle. Since the radius is \( r \), the diameter \( AB \) is:
\[ AB = 2r \]
The height of the triangle is the perpendicular distance from the apex \( C \) of the triangle to the base \( AB \). This height is equal to the radius \( r \) of the semicircle.

### Step 3: Calculate the Area of the Triangle
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 found:
\[ A = \frac{1}{2} \times (2r) \times r \]
\[ A = \frac{1}{2} \times 2r^2 \]
\[ A = r^2 \]

### Conclusion
Thus, the area of the largest triangle that can be inscribed in a semicircle of radius \( r \) is:
\[ \boxed{r^2} \]