Answer:
C. 4 cm²
Explanation:
You want the maximum possible area of a quadrilateral inscribed in a semicircle.
Area
The figure is symmetrical about a vertical line, so the area will be maximized when the area of half the quadrilateral is maximized. The area of the quadrilateral in the right half of the figure is the product of the x- and y-coordinates of the corner point on the circle.
For coordinates (x, y), the area is A=xy. The graph of this function is a hyperbola, symmetric about the line y=x. The larger the value of A, the farther the graph is from the origin.
The maximum possible value of A will be found where the graph of xy=A is tangent to the circle. That point of tangency will lie on the circle and on the line y = x.
Corner point
The equation for the semicircle is ...
x² +y² = 2²
When x=y, this is ...
x² +x² = 4
x² = 2
This is the area of the quadrilateral in the right half of the figure.
The entire quadrilateral has an area twice this, or 2·2 = 4 (square cm).
The maximum possible area of the quadrilateral is 4 cm².
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Additional comment
You will notice the figure is half of a figure of a whole circle with a square inscribed. This is not a coincidence.
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