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In this section, we explore the crucial relationship between the center of a circle, a chord, and the perpendicular that is drawn to the chord. Theorems related to this relationship highlight that such a perpendicular not only bisects the chord but also offers insights into distance properties related to equal chords. Examples and classroom discussions illustrate these concepts effectively.
This section explores the Perpendicular from the Centre to a Chord in a circle and highlights two important theorems:
An activity connecting equal chords and their distances from the center leads to Theorem 9.5, affirming that equal chords are equidistant from the center. Finally, the converse of this theorem, Theorem 9.6, establishes that chords equidistant from the center are equal in length.
These theorems and their corresponding proofs illuminate foundational properties of circles and chords, and they emphasize the significant relationship between angles, distances, and lengths in circular geometry.
Perpendicular bisector to a chord: The perpendicular drawn from the center of the circle to a chord bisects that chord.
Equidistance of chords: Equal chords in a circle are equidistant from the center.
In a circle so round and wide, the centerβs line will always guide. Perpendiculars hold a hidden key, to bisect chords perfectly!
Once upon a circle, where a wise old center knew that every chord needed its partner bisected; together they danced the elegance of distance and equal length.
Remember 'CE': Chords are Equal, when they share the same Distance from the center.
{'example': 'Given a circle with center O and a chord AB, if OM is perpendicular to AB, prove that AM = MB.', 'solution': 'Since OM is perpendicular, triangles OMA and OMB are congruent by the Hypotenuse-Leg theorem. Therefore, AM = MB.'}
{'example': 'If chords AB and CD are equal and both are equidistant from the center, prove they are equal.', 'solution': 'Since they are equidistant from the center, both will have the same length by Theorem 9.6.'}
Term: Chord
Definition: A line segment whose endpoints lie on a circle.
A line segment whose endpoints lie on a circle.
Term: Perpendicular
Definition: A line meeting another line at a right angle (90 degrees).
A line meeting another line at a right angle (90 degrees).
Term: Bisect
Definition: To divide into two equal parts.
To divide into two equal parts.
Term: Equidistant
Definition: At equal distances from a given point.
At equal distances from a given point.