Perpendicular Chord Bisector Theorem

Consider the segments $\overline{O C}$ and $\overline{O D} .$ Since $\overline{A B}$ and $\overline{C D}$ are perpendicular segments, then $∠ O E C$ and $∠ O E D$ are right angles. Therefore, $△ O C E$ and $△ O D E$ are right triangles.

Next, since two radii of a circle are congruent, then

\overline{O C}

and

\overline{O D}

are congruent. Furthermore, by the Reflexive Property of Congruence,

\overline{O E}

is congruent to itself.

\overline{O C} ≅ \overline{O D} and \overline{O E} ≅ \overline{O E}

Therefore, the hypotenuse and one leg of

△ O C E

are congruent to the hypotenuse and the corresponding leg of

△ O D E .

By the Hypotenuse Leg Theorem,

△ O C E

and

△ O D E

are congruent triangles.

△ O C E ≅ △ O D E

Since corresponding parts of congruent triangles are congruent, then the other legs of the triangles are also congruent.

$\overline{E C} ≅ \overline{E D}$

The proof for the first part of the statement has been completed.

Now, to show the congruence of the arcs

B C

and

B D,

the properties of congruent right triangles will be considered. Since corresponding parts of congruent triangles are congruent, it can be said that

∠ C O E

and

∠ D O E

are congruent angles.

∠ C O E ≅ ∠ D O E

Therefore, by the Congruent Central Angles Theorem,

B C

and

B D

are congruent.

$B C ≅ B D$

This completes the proof.

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Perpendicular Chord Bisector Theorem

Proof

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