Problem:
Distinct lines and lie in the -plane. They intersect at the origin. Point is reflected about line to point , and then is reflected about line to point . The equation of line is , and the coordinates of are . What is the equation of line ?
Answer Choices:
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Solution:
It is well known that the composition of 2 reflections, one after another, about two lines and , respectively, that meet at an angle is a rotation by around the intersection of and .
Now, we note that is a degree rotation clockwise of about the origin, which is also where and intersect. So is a degree rotation of about the origin clockwise.
To rotate degrees clockwise, we build a square with adjacent vertices and . The other two vertices are at and . The center of the square is at , which is the midpoint of and . The line passes through the origin and the center of the square we built, namely at and . Thus the line is . The answer is .
The problems on this page are the property of the MAA's American Mathematics Competitions