Course:MATH102/Question Challenge/1998 December Q3
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Question
Find the area of the largest rectangle that can be inscribed in the semicircle of radius .
Hints
Hint 1 

Use the coordinates to label the point on the circle at the corner of the rectangle. What is the relationship between , , and ? This relationship will be a constraint in this optimization problem. 
Hint 2 

What is the width and height of the rectangle in terms of and ? 
Hint 3 

Write down the area of the rectangle in terms of and , then use the constraint to eliminate either or . 
Solutions
Solution 

Let us try to solve the problem geometrically. The radius of the semicircle forms two right triangles within the rectangle, one on the rectangle's left and one on its right. Due to symmetry, there are also another two congruent triangles mirrored along hypotenuse of both right triangles. Together these four right triangles make up the total area of the rectangle inscribed within the semicircle. When we rearrange these right triangles we see they form a square with sidelength . The equation for the area of the square and therefore the area of the rectangle inscribed by the semicircle is therefore:
