Tiffany N. KeysŐ Assignment 4:
Concurrency and Perpendicular Bisectors
INVESTIGATION: Prove that the three perpendicular bisectors of the sides of a triangle are concurrent.
Given triangle ABC, construct the midpoint, M, of AB.
Construct the perpendicular bisector, x, of AB.
Construct a point, D, on x, then construct DA and DB
Since AM = BM, angle(AMD) = angle(BMD).
Since triangle(AMD) = triangle(BMD) by the Side-Angle-Side Theorem. Therefore, AD = DB.
Construct the perpendicular bisector, y, of BC.
Since AB and BC are not parallel, lines x and y must intersect.
Merge point D to the point of intersection for lines x and y.
Therefore, CF = FB, angle (CFD) = angle(BFD), triangle (CFD) = triangle (BFD), and CD = DB.
By the Axiom that states that every segment is congruent to itself, we know that if CD=DB and DB=DA, then DA=DC.