9.3 Equal Chords and their Distances from the Centre

Description

Quick Overview

This section explores the relationship between equal chords in a circle and their distances from the center, detailing several key theorems and proofs.

Standard

In this section, students learn that equal chords of a circle are equidistant from the center, and vice versa. Theorems are presented to support this, including proofs of related properties of chords and distances. Classroom activities reinforce these concepts.

Detailed

Detailed Summary

This section discusses the significant properties associated with equal chords in a circle and their distances from the center. One of the fundamental definitions introduced is that the distance from a point to a line is defined as the length of the perpendicular drawn from the point to the line. In the context of circles, it is observed that longer chords are closer to the center while shorter chords are farther away.

The section emphasizes two essential theorems:

  1. Theorem 9.5: Equal chords of a circle (or congruent circles) are equidistant from the center (or centers).
  2. Theorem 9.6: Chords equidistant from the center of a circle are equal in length.

To support students' understanding, it describes several activities that involve drawing equal chords, measuring distances, and using tracing paper to visualize and verify these relationships. By engaging in these exercises, students observe how the properties hold true, consolidating their grasp of the theorems. The section culminates with an example that demonstrates the application of the concepts learned.

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Key Concepts

  • Chord: A line segment connecting two points on a circle.

  • Distance: The shortest length from a point to a line, identified as the perpendicular.

  • Equal Chords: If two chords are equal in length, they are equidistant from the center of a circle.

Memory Aids

🎵 Rhymes Time

  • Chords that are equal, distances the same, Measure them close in this circle game.

📖 Fascinating Stories

  • Imagine two friends, equal in height, Standing far apart, aligned in sight. To measure their distance from the center's light, They find they're the same, what a wonderful sight!

🧠 Other Memory Gems

  • EQUAL = Equal chords are Equidistant from the Center

🎯 Super Acronyms

EDD

  • Equal Distance
  • Equal Chord.

Examples

  • {'example': 'If two intersecting chords of a circle make equal angles with the diameter passing through their intersection point, prove that the chords are equal.', 'solution': 'Let the chords be AB and CD intersecting at point E. By drawing perpendiculars to the chords from the center O, it can be shown that due to symmetry and equality of angles, OL = OM which leads to AB = CD.'}

Glossary of Terms

  • Term: Chord

    Definition:

    A line segment with both endpoints on a circle.

  • Term: Equidistant

    Definition:

    Being at equal distances from a common point, in this case, the circle's center.

  • Term: Perpendicular

    Definition:

    A line that makes a right angle (90 degrees) with another line or surface.