9.2 Perpendicular from the Centre to a Chord

Description

Quick Overview

The section discusses the properties of the perpendicular drawn from the center of a circle to a chord, including its ability to bisect the chord and its relation to the concept of distance from the center.

Standard

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.

Detailed

Detailed Summary

This section explores the Perpendicular from the Centre to a Chord in a circle and highlights two important theorems:

  1. Theorem 9.3 states that the perpendicular from the center of a circle to a chord bisects the chord. This theorem is demonstrated through a folding activity, where students see that when the chord overlaps, the two segments created by the perpendicular meet at the midpoint, affirming that the lengths are equal.
  2. Theorem 9.4 is the converse of Theorem 9.3, stating that if a line from the center bisects a chord, it is perpendicular to that chord.

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.

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Key Concepts

  • 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.

Memory Aids

🎵 Rhymes Time

  • In a circle so round and wide, the center’s line will always guide. Perpendiculars hold a hidden key, to bisect chords perfectly!

📖 Fascinating Stories

  • 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.

🧠 Other Memory Gems

  • Remember 'CE': Chords are Equal, when they share the same Distance from the center.

🎯 Super Acronyms

Use 'PB' for Perpendicular Bisection to remember the key theorem!

Examples

  • {'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.'}

Glossary of Terms

  • Term: Chord

    Definition:

    A line segment whose endpoints lie on a circle.

  • Term: Perpendicular

    Definition:

    A line meeting another line at a right angle (90 degrees).

  • Term: Bisect

    Definition:

    To divide into two equal parts.

  • Term: Equidistant

    Definition:

    At equal distances from a given point.