The area of the largest triangle that can be inscribed in a semi-circle of radius $r$, is

### Concept & Inscribed Triangle Geometry Any triangle inscribed in a semi-circle with its base on the diameter will be a right-angled triangle (Thales's Theorem). The area of a triangle is $\frac{1}{2} \times \text{base} \times \text{height}$. To maximize the area for a fixed base, the height must be maximized. ### Step-by-Step Solution 1. **Determine the Base:** The largest triangle inscribed in a semi-circle must span the entire flat edge of the semi-circle. Therefore, its base is the diameter of the semi-circle. $\text{Base} = 2r$ 2. **Determine the Maximum Height:** The third vertex must lie on the semi-circle arc. To maximize the height of the triangle, this vertex must be at the highest point of the semi-circle, which is directly above the center. The distance from the center to this highest point is simply the radius. $\text{Maximum Height} = r$ 3. **Calculate the Maximum Area:** $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}$ $\text{Area} = \frac{1}{2} \times (2r) \times (r)$ $\text{Area} = r^2$ ### Exam Strategy & Shortcut Visualize the semi-circle. The base is locked at $2r$. The highest you can drag the peak of the triangle is the top center, which is height $r$. Half of $2r \times r$ is just $r^2$. The visualization is instant. ### Common Pitfall A common mistake is assuming the formula for the area of the semi-circle ($\frac{\pi r^2}{2}$) is needed, or mistakenly picking $2r^2$ by forgetting to multiply by the $1/2$ in the triangle area formula. ### Final Answer Therefore, the correct answer is **$r^2$**.