Problem:
Points and are vertices of with . The altitude from meets the opposite side at . What are the coordinates of point ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The altitude lies on a line of symmetry for the isosceles triangle. Under reflection about this line, will be sent to . Because is obtained from by adding to the -coordinate and subtracting from the -coordinate, is obtained from by subtracting from the -coordinate and adding to the -coordinate. Thus the third vertex has coordinates .
To find the coordinates of , note that is the midpoint of . Therefore
Solving these equations gives and , so .
The problems on this page are the property of the MAA's American Mathematics Competitions