Problem:
Triangle lies in the first quadrant. Points , and are reflected across the line to points , and , respectively. Assume that none of the vertices of the triangle lie on the line . Which of the following statements is always true?
Answer Choices:
A. Triangle lies in the first quadrant.
B. Triangles and have the same area.
C. The slope of line is .
D. The slopes of lines and are the same.
E. Lines and are perpendicular to each other.
Solution:
The reflection of the point across the line is the point . Because the coordinates of the points in the original triangle are all positive, it follows that the coordinates of the images will also be all positive. Thus is always true. It is a property of reflections that the line segment connecting a point not on the line of reflection and its image is perpendicular to the line of reflection. This fact shows that and are always true. Reflection is a rigid transformation, and therefore areas are preserved, so is always true. The statement is not true in general. As an example, consider a triangle such that the side is parallel to the line . Then the side in the image will also be parallel to , which shows that lines and are not perpendicular. Thus the statement is not always true. In fact, because reflection across the line interchanges the roles of and , the slope of a nonvertical/non-horizontal line and the slope of its image are reciprocals, not negative reciprocals. A line will be perpendicular to its reflection across the line if and only if the line is horizontal or vertical, in which case its image will be vertical or horizontal, respectively.
The problems on this page are the property of the MAA's American Mathematics Competitions