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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.
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:
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.
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.
Chords that are equal, distances the same, Measure them close in this circle game.
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!
EQUAL = Equal chords are Equidistant from the Center
{'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.'}
Term: Chord
Definition: A line segment with both endpoints on a circle.
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.
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.
A line that makes a right angle (90 degrees) with another line or surface.