Problem of the Week #31: Monday August 7th, 2023 As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
Let’s start with a right triangle with a height of 1 unit and consider two ways of dividing it into two triangles. The first is to bisect the base, like this:
The second is to bisect the angle opposite to the base, like this:
Notice that in both of these cases, the dividing line intersects the base.
a). In which case is the point of intersection closer to the right angle?
b). If the base of the triangle is also 1 unit long, how far apart are these two intersection points?
c). Do the intersection points become closer or further apart as we increase the length of the base?
d). What is the maximum proportion of this difference compared to the whole base for all right triangles in this position that have a height of 1 unit?
e). Do the intersection points become closer or further apart as we decrease the measurement of the right angle (at the bottom left)?
f). What is the maximum proportion of this difference compared to the whole base for all triangles (not necessarily right triangles) in this position and have both a base and a height of 1 unit?
g). Share your own problem inspired by this one.
h). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.