InterMath | Workshop Support | Write Up Template

Title
SAM

Problem Statement
How many altitudes and medians are there in a triangle? In what type(s) of triangles is one altitude the same segment as a median? How many symmetry (folding or reflecting) lines can you draw in this type of triangle?

In what type(s) of triangles are all of the corresponding altitudes and medians the same? How many symmetry lines can you draw in this type of triangle? Can exactly 2 of the altitudes of a triangle be the same segment as 2 medians? Why or why not?

Problem setup
Discover the relationship between symmetry lines and triangles through the exploration of there respective altitudes and medians. This problem is similar to others we do in class because we learn and understand through manipulations. It is also similar to the problems we worked with trapezoids in that it deals with the concept of altitude.

Plans to Solve/Investigate the Problem

First I will begin by visualizing the different types of triangles, and make sure I understand the concepts of altitude and median. I will then use Geometer’s sketchpad to construct the different types of triangles as well as there altitudes and medians. Finally, I will make conjectures about the relationship between symmetry and altitude and medians using the constructions.

Investigation/Exploration of the Problem
To solve this problem I will first try and visualize the different types of triangles and their altitudes and medians as I try to recall my understanding of altitude and median. After visualizing I will begin to construct and measure the properties of the various types of triangles using Sketch pad including equilateral, right, obtuse and acute isosceles, and obtuse and acute scalene. After the constructions are complete I will construct all three medians by connecting the vertices and opposite mid points on each triangle. I will also construct the as many of the altitudes are possible for each triangle by constructing a perpendicular line passing through a vertex and its opposite mid point. The lines that are both an altitude and a median are marked with a special color, in this case gray. Through the investigation of my constructions, I determined that all triangles had three altitudes and three medians, even though some of the altitudes did not pass through the opposite side. I also determined that there were three types if triangles where one median was the same as one altitude including right, acute and obtuse isosceles triangles. These triangles had line symmetry about that line, but I saw no evidence of rotational symmetry. The equilateral triangle was the only type of triangle where all three altitudes and medians the same. I also determined that the equilateral had three points of line symmetry and also rotational symmetry. I also determined that no triangle could have exactly two altitude and two median lines the same. There can only be one or three altitude and median lines the same never exactly two, because if there are two that would mean all three would be and you would never get exactly two.

Extensions of the Problem
Discuss the relationship between the number of symmetry lines in a triangle using altitudes and medians in your description.

Author & Contact
Nathan Little
nlittle@aug.edu

Link(s) to resources, references, lesson plans, and/or other materials
My constructions
Intermath

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