Circles

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Definition of a Circle

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A circle is a set of all points in a plane that are at a given distance (radius) from a given point (center).

Terms Related to a Circle

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• Radius: The distance from the center of a circle to any point on its circumference.
• Diameter: The longest chord of a circle, passing through its center.
• Chord: A line segment joining two points on the circumference of a circle.
• Secant: A line that intersects the circle at two points.
• Tangent: A line that intersects the circle at exactly one point.

Key Formulas

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• Circumference of a Circle: C = 2πr
• Area of a Circle: A = πr^2
• Area of a Sector: A = (θ/360) × πr^2, where θ is the angle subtended by the sector at the center.
• Length of an Arc: l = (θ/360) × 2πr, where θ is the angle subtended by the arc at the center.
• Equation of a Circle with Center (a, b) and Radius r: (x - a)^2 + (y - b)^2 = r^2

Theorems

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• Thales' Theorem: The angle subtended by a diameter of a circle at its circumference is a right angle (90°).
• Angle in a Semicircle: The angle subtended by a diameter of a circle at its circumference is 90°.
• Angle in a Circle: The angle subtended by a chord at the center of a circle is twice the angle subtended by the same chord at the circumference.
• Tangent-Chord Theorem: The angle between a tangent and a chord is equal to the angle in the alternate segment.

Important Results

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• Pythagoras' Theorem in a Circle: In a right-angled triangle inscribed in a circle, the hypotenuse is the diameter of the circle.
• Circle and Triangle Properties: A triangle inscribed in a circle must have its circumcenter at the center of the circle.

Key Points to Remember

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• A circle is defined as the set of all points at a given distance from a given point (center).
• The circumference, area, and sector area formulas are essential.
• Thales' Theorem and the Angle in a Semicircle are key theorems to remember.
• The tangent-chord theorem is also important.