mathleaks.com mathleaks.com Start chapters home Start History history History expand_more
{{ item.displayTitle }}
navigate_next
No history yet!
Progress & Statistics equalizer Progress expand_more
Student
navigate_next
Teacher
navigate_next
Expand menu menu_open Minimize
{{ filterOption.label }}
{{ item.displayTitle }}
{{ item.subject.displayTitle }}
arrow_forward
No results
{{ searchError }}
search
menu_open home
{{ courseTrack.displayTitle }}
{{ statistics.percent }}% Sign in to view progress
{{ printedBook.courseTrack.name }} {{ printedBook.name }}
search Use offline Tools apps
Login account_circle menu_open

Perpendicular Chord Bisector Theorem

Rule

Perpendicular Chord Bisector Theorem

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

Image not found. We apologize, please report this so that we can fix it as soon as possible!File = mljsx_Rules_Perpendicular_Chord_Bisector_Theorem_1.svg, id = Rules_Perpendicular_Chord_Bisector_Theorem_1

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

and $\ \Arc{BC} \cong \Arc{BD}$

Proof

Consider the segments and Since and are perpendicular segments, then and are right angles. Therefore, and are right triangles

Image not found. We apologize, please report this so that we can fix it as soon as possible!File = mljsx_Rules_Perpendicular_Chord_Bisector_Theorem_2.svg, id = Rules_Perpendicular_Chord_Bisector_Theorem_2

Next, since two radii of a circle are congruent, then and are congruent. Furthermore, by the Reflexive Property of Congruence, is congruent to itself. Therefore, the hypotenuse and one leg of are congruent to the hypotenuse and the corresponding leg of

Image not found. We apologize, please report this so that we can fix it as soon as possible!File = mljsx_Rules_Perpendicular_Chord_Bisector_Theorem_3.svg, id = Rules_Perpendicular_Chord_Bisector_Theorem_3

By the Hypotenuse Leg Theorem, and are congruent triangles. Since corresponding parts of congruent triangles are congruent, then the other legs of the triangles are also congruent.

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

Image not found. We apologize, please report this so that we can fix it as soon as possible!File = mljsx_Rules_Perpendicular_Chord_Bisector_Theorem_4.svg, id = Rules_Perpendicular_Chord_Bisector_Theorem_4

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 and are congruent angles. Therefore, by the Congruent Central Angles Theorem, and are congruent.

This completes the proof.

Image not found. We apologize, please report this so that we can fix it as soon as possible!File = mljsx_Rules_Perpendicular_Chord_Bisector_Theorem_5.svg, id = Rules_Perpendicular_Chord_Bisector_Theorem_5