Problem:
Points and are the midpoints of sides and of . As moves along a line that is parallel to side , how many of the four quantities listed below change?
the length of the segment
the perimeter of
the area of
the area of trapezoid
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Since is similar to and , it follows that . Since the base and the altitude to of do not change, the area does not change. The altitude of the trapezoid is half that of the triangle, and the bases do not change as changes, so the area of the trapezoid does not change. Only the perimeter changes (reaching a minimum when is isosceles).
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions