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The summary details fundamental theorems about circles, such as the relations between equal chords, their subtended angles at the center, and properties of cyclic quadrilaterals. It emphasizes the significance of angles subtended by arcs, congruent arcs, and the relationship to cyclic shapes.
In this chapter, you have studied various properties and theorems related to circles, particularly focusing on chords and angles. A circle is defined as a collection of all points in a plane equidistant from a fixed point. Key points highlighted in this summary include:
These key points form a foundational understanding of the relationships in circles that are essential for deeper exploration of geometric properties.
Equal Chords: Chords of equal length subtend equal angles at the center.
Perpendicular Bisector: A perpendicular line from the center to a chord bisects the chord.
Equidistant Chords: Equal chords are at equal distances from the center.
Arc and Chord Correspondence: Congruent arcs imply equal chords, and vice versa.
Cyclic Quadrilaterals: The sum of opposite angles in cyclic quadrilaterals is 180Β°.
Chords that are equal, angles will match,
Imagine two friends (chords) on a merry-go-round. When they travel the same distance (equal), they both see the same view (equal angles) from the center.
Remember 'EQA' - Equal chords mean Equal angles.
{'example': 'If two equal chords are drawn in a circle, prove the angles subtended at the center are equal.', 'solution': 'If AB = CD, then β AOB = β COD.'}
{'example': 'Two chords AB and CD are at equal distances from the center, prove they are equal.', 'solution': 'If OM = ON, then AB = CD.'}
Term: Chord
Definition: A line segment with both endpoints on a circle.
A line segment with both endpoints on a circle.
Term: Circle
Definition: A set of all points in a plane that are a fixed distance from a center point.
A set of all points in a plane that are a fixed distance from a center point.
Term: Cyclic Quadrilateral
Definition: A quadrilateral whose vertices lie on the circumference of a circle.
A quadrilateral whose vertices lie on the circumference of a circle.
Term: Angle Subtended
Definition: The angle formed between two lines drawn from the ends of a chord to a point.
The angle formed between two lines drawn from the ends of a chord to a point.
Term: Arc
Definition: A portion of a circle defined by two endpoints on the circle.
A portion of a circle defined by two endpoints on the circle.