The diagram below shows a Christmas Tree. As it happens, the shape of the tree (an
equilateral triangle with a segment at the bottom) makes it possible to draw a circle around
the tree as shown.
How tall is the trunk of the tree (the segment) in relation to the entire tree? Explain.
Solution to the Problem:
The trunk is 1/4 the height of the entire tree.Draw the perpendicular bisectors of the three sides, which are also the angle bisectors since it is an equilateral triangle. Six congruent right triangles are formed and each is a 30-60-90 triangle. The bisectors meet at the center. Each shorter segment (the side opposite the 30 degree angle is 1/2 of the hypotenuse). Since the the centroid (the intersection of the medians) is (2/3) the distance from a vertex to the midpoint of the opposite side, then the segment or "trunk" of the tree must equal half the radius, or one-fourth of the total height of the tree.
Correctly solved by:
| 1. James Alarie | Flint, Michigan |
| 2. Chad Fore | Gate City, Virginia |
| 3. Brooks Garris |
Lake View High School, Lake View, South Carolina |
| 4. Keith Mealy | Cincinnati, Ohio |
| 5. Harlan Benedict |
Mountain View High School, Mountain View, Wyoming |