Introduction to Circles

Introduction to Circles

• A circle is a set of all points in a plane that are equidistant from a fixed point called the center.
• The fixed distance from the center to any point on the circle is called the radius.
• A line segment joining any two points on a circle is called a chord.
• A chord that passes through the center of the circle is called a diameter. The diameter is twice the radius.
• A line that intersects a circle at exactly one point is called a tangent.
• The point where a tangent touches the circle is called the point of tangency.
• A sector of a circle is the region bounded by two radii and the intercepted arc.
• A segment of a circle is the region bounded by a chord and the intercepted arc.

Properties of Circles

• All radii of a circle are equal.
• The diameter of a circle is twice its radius.
• A circle has 360 degrees.
• The angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the remaining part of the circle.
• A tangent to a circle is perpendicular to the radius drawn to the point of contact.
• Equal chords of a circle subtend equal angles at the center. Conversely, if two chords of a circle subtend equal angles at the center, they are equal.
• The perpendicular from the center to a chord bisects the chord.

Tangents to a circle

• A tangent to a circle is a line that intersects the circle at exactly one point.
• The point where a tangent touches the circle is called the point of contact.
• A tangent to a circle is perpendicular to the radius drawn to the point of contact.

Theorems on Chords, Arcs, and Angles

• Equal chords subtend equal angles at the center.
• If two chords are equidistant from the center, they are equal.
• The angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the remaining part of the circle.

Cyclic Quadrilateral

• A quadrilateral whose vertices lie on a circle is called a cyclic quadrilateral.
• The sum of opposite angles of a cyclic quadrilateral is 180 degrees.
• Conversely, if the sum of any pair of opposite angles of a quadrilateral is 180 degrees, the quadrilateral is cyclic.

Angle Subtended by an Arc of a Circle

• The angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the remaining part of the circle.

Constructions

• Construct a circle passing through three given non-collinear points.

Measurements of Angles and Arcs

• Understanding of central, inscribed, exterior and interior angles.
• Relationship between different angles and arcs in a circle.
• Identifying different types of arcs and recognizing their properties.
• Calculating central, inscribed angles, and the degree measure of arcs.

Important Concepts to Remembe

• Understanding definitions: radius, diameter, chord, tangent, sector, segment.
• Knowing the properties of circles and the theorems associated with them.
• Application of these properties to solve problems.
• Practicing questions of varying difficulty.

This quiz covers the fundamental concepts and properties of circles. You will learn about key terms such as radius, diameter, chord, tangent, sector, and segment. Test your understanding of the relationships and properties that define circles in geometry.