Discussion
The area of the largest triangle that can be inscribed in a semicircle of radius x (in square units) is :
Answer: B
The largest area is \(x^{2}\) square units.
The largest triangle in a semicircle of radius \(x\) takes the diameter as its base, of length \(2x\), and its third vertex at the top of the arc, at the greatest possible height \(x\):
\[ \text{Area} = \tfrac{1}{2} \times 2x \times x = x^{2} \]
\(\tfrac{1}{2}x^{2}\) is what a triangle with base \(x\) would give, and \(2x^{2}\) and \(4x^{2}\) forget the factor of one half or double the height; none can fit inside the semicircle, whose own area is only \(\tfrac{\pi}{2}x^{2}\), about \(1.57x^{2}\).
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