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.
