Perpendicular Bisectors of Triangles

Objective: To use perpendicular bisectors of sides of triangles to prove identities of particular triangles (isosceles, equilateral, right).

Instructions: Play around with the triangle. Move the vertices in order to see what kind of triangle is produced when each of the following is done. Note that the vertex A is fixed at (0,0).

Questions:
1. Move vertex C so that it lies on the y-axis. Then move vertex B so that it lies on the perpendicular bisector of side b.
a. By noticing the length of sides c and a at this position, what kind of triangle is this?
b. Why do you suppose this is so?
c. How can you show algebraically that the two sides, a and c, are equal? (Hint: Give vertices A, B, and C arbitrary coordinates.)
2. Now move vertex B so that it lies on the x-axis and vertex C to lie on the y-axis.
a. What kind of triangle is this?
b. What do you notice about the point D (which is the intersection of the perpendicular bisectors)?
c. Move vertices B and C so that the length of side c is equal to the length of side b. What do you notice about the four triangles that are created inside triangle ABC? Prove that they are all congruent to each other.
3. How can you arrange the vertices and perpendicular bisectors to configure an equilateral triangle?
a. Once you have configured this equilateral triangle, prove that all 6 triangles inside of triangle ABC are congruent to one another.

Allison Noble, Created with GeoGebra