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{{ printedBook.courseTrack.name }} {{ printedBook.name }} If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

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In the diagram above, $AB$ is the diameter and $CD$ is a chord such that $AB$ is perpendicular to $CD.$ Therefore, the following congruences hold true.

$EC≅ED$ and $\ \Arc{BC} \cong \Arc{BD}$

Consider the segments $OC$ and $OD.$ Since $AB$ and $CD$ are perpendicular segments, then $∠OEC$ and $∠OED$ are right angles. Therefore, $△OCE$ and $△ODE$ are right triangles

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Next, since two radii of a circle are congruent, then $OC$ and $OD$ are congruent. Furthermore, by the Reflexive Property of Congruence, $OE$ is congruent to itself. $OC≅ODandOE≅OE $ Therefore, the hypotenuse and one leg of $△OCE$ are congruent to the hypotenuse and the corresponding leg of $△ODE.$

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By the Hypotenuse Leg Theorem, $△OCE$ and $△ODE$ are congruent triangles. $△OCE≅△ODE $ Since corresponding parts of congruent triangles are congruent, then the other legs of the triangles are also congruent.

$EC≅ED$

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

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Now, to show the congruence of the arcs $\Arc{BC}$ and $\Arc{BD},$ the properties of congruent right triangles will be considered. Since corresponding parts of congruent triangles are congruent, it can be said that $∠COE$ and $∠DOE$ are congruent angles. $∠COE≅∠DOE $ Therefore, by the Congruent Central Angles Theorem, $BC$ and $BD$ are congruent.

$BC≅BD$

This completes the proof.

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