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
$$\mathtt{\color{blue}{\Delta ACD} \cong \color{orange}{\Delta BCD}}$$ 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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