Overview
This lecture covers the fundamental terminology of circles and key theorems about angles, chords, tangents, and cyclic quadrilaterals, essential for understanding circle geometry.
Basic Circle Terminology
- The center is the point equidistant from all points on the circle.
- A radius is a line from the center to any point on the circle; all radii are equal.
- A diameter is a line passing through the center, twice the radius.
- A chord connects any two points on the circle without passing through the center; the longest chord is the diameter.
- An arc is a curved part of the circle between two points.
- A sector is the area between two radii and an arc.
- A segment is a region formed when a chord divides the circle.
- The circumference is the total boundary length of the circle.
- A tangent touches the circle at exactly one point, without crossing it.
- A secant intersects the circle at two points.
- "Subtended" describes the angle created at a point by an arc or chord.
Key Circle Theorems
- Theorem 1: Angles subtended by the same arc at the circumference are equal.
- Theorem 2: The angle subtended at the center is twice the angle at the circumference by the same arc.
- Theorem 3: The angle in a semicircle is always 90°.
- Theorem 4: Opposite angles of a cyclic quadrilateral (all vertices on the circle) are supplementary (sum to 180°).
- Theorem 5: A tangent is perpendicular to the radius at the point of contact.
- Theorem 6: Tangents drawn from an external point to a circle are equal in length; the line to the center bisects the angle between tangents.
- Theorem 7: The angle between a tangent and a chord from the contact point equals the angle in the opposite segment by that chord.
- Theorem 8: A line that is perpendicular to and bisects a chord passes through the circle’s center.
- Theorem 9: Equal chords subtend equal angles at the center and at the circumference.
- Theorem 10: The product of the segments of two secants from the same external point is equal; with a tangent: PA × PB = PC² (tangent-secant theorem).
- Theorem 11: If two chords intersect inside a circle, the products of their segments are equal.
Key Terms & Definitions
- Radius — Line from circle’s center to any point on the circle.
- Diameter — Line passing through the center; twice the radius.
- Chord — Line joining any two points on the circle.
- Arc — Curved segment of the circle between two points.
- Sector — Area between two radii and their connecting arc.
- Segment — Region between a chord and an arc.
- Tangent — Line touching the circle at only one point.
- Secant — Line intersecting the circle at two points.
- Cyclic Quadrilateral — Quadrilateral with all vertices on the circle.
Action Items / Next Steps
- Solve practice problems using these theorems.
- Review definitions and theorems for better recall.
- Try solving the example problems posed in the lecture.