Perpendicular Bisector Theorem

Press on the numbers to see the steps of the proof.

1 Draw \(\mathtt{\overline{AB}}\). Then construct the line which bisects \(\mathtt{\overline{AB}}\) at a right angle (i.e., the perpendicular bisector).

2 Draw a point on the perpendicular bisector above or below \(\mathtt{\overline{AB}}\). Connect this point to the endpoints of \(\mathtt{\overline{AB}}\).

3 \(\mathtt{m\overline{AC} = m\overline{CB}}\), because the perpendicular bisector bisects \(\mathtt{\overline{AB}}\), and \(\mathtt{m\overline{CD} = m\overline{CD}}\), because of the Reflexive Property

4

Thus, \(\mathtt{\overline{AD} \cong \overline{BD}}\), because of \(\mathtt{\color{blue}{\Delta ACD} \cong \color{orange}{\Delta BCD}}\) by Side-Angle-Side congruence.Thus, \(\mathtt{\overline{AD} \cong \overline{BD}}\), because of CPCTC

5 Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.

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