Assignment 2

Write clearly the solutions to the following problems, proofs and constructions and turn them in. Note that proofs can use the theorems we have proved in class.

2.1 Concurrence of perpendicular bisectors of a triangle

Draw a general* triangle ABC and construct with straightedge and compass the perpendicular bisectors of the sides.

(a) Prove that the 3 perpendicular bisectors are concurrent at a point O for any triangle ABC.

(b) Construct the circle with center O through A.

(c) Explain why the circle with center O through A also passes through B and through C for any triangle ABC.

*General is a somewhat ill-defined term, but referring to an example, it means not special, so it should not be isosceles, equilateral, having a right angle or any other "special" angle such as 60 degrees. It means what one might call a "random" triangle if one could give a good definition of random.

2.2. SSA for triangles

Suppose that ABC and DEF are triangles, with AB = DE, AC = DF and angle B = angle E. This condition is called SSA (or you can reverse the order of letters if you like!).

Construct an example that shows that SSA is not a valid criterion for triangle congruence. Specifically, find two lengths AB and AC, and an angle B and then construct two triangles A'B'C' and A"B"C" that satisfy the SSA criterioin but are not congruent.