Problem:
Figure is a square. Point is the origin, and point has coordinates . What are the coordinates for so that the area of triangle equals the area of square
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Solution:
Since is a square and point has coordinates , it follows that point has coordinates and point has coordinates . Thus the side of square is 2 and the area of the by square is . The area of triangle is . Since the area of the triangle equals the area of the square, . Thus the point has coordinates .
If the area of square and triangle are equal, then the area of the small triangle with side that is in the square but not in the triangle must equal the area of the small triangle with side that is in the triangle but not in the square. These two triangles will be the same size and shape when intersects at its midpoint and , so must have coordinates .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions