Answer:
9 cm²
Step-by-step explanation:
Sometimes these inscribed polygon problems respond nicely to a little geometrical thinking. The figure is symmetrical about the vertical line through the center of the square(s).
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Analysis
That line of symmetry divides the figure into halves that are twice as tall as wide. That same ratio applies to both the outer square and the inner (blue shaded) square.
In other words, a line from the center of the top segment through either bottom corner of the figure will pass through the corners of both squares. That line will have a slope of ±2, making it easy to find the coordinates of one of the lower corners of the shaded square.
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System of Equations
Analytically, we can write the equation of the right circle shown in the attachment as ...
x² +y² = 5²
and the equation of the line as ...
y = 2x +5
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Solution of Equations
Then the x-value of the lower left corner will be found where these intersect.
x² +(2x +5)² = 25 . . . . . . substitute for y
x² +4x² +20x +25 25 . . . . eliminate parentheses
5x² +20x = 0 . . . . . . . . . . . . subtract 25
5x(x +4) = 0 . . . . . . factor
x = -4 or 0 . . . . . the x-coordinates of the points of intersection
The y-coordinate of the lower left point is ...
y = 2(-4) +5 = -3
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Area of the Square
Since the top edge of the blue square has a y-coordinate of 0, this means the square is 3 cm on a side. Its area is ...
A = s² = (3 cm)² = 9 cm² . . . . area of the blue shaded square
