Overview
This lecture introduces circle theorems, focusing on key properties and relationships of angles, chords, tangents, and shapes within circles relevant for exams.
Chords and Their Properties
- A chord is a line segment joining two points on a circle, not necessarily passing through the center.
- The perpendicular bisector of a chord always passes through the center of the circle.
Angles in Semicircles
- Any triangle drawn inside a semicircle with its base as the diameter creates a right angle (90°) at the opposite vertex.
Tangents to Circles
- A tangent is a line touching the circle at exactly one point.
- The angle between a tangent and the radius at the point of contact is always 90°.
- The lengths of tangents drawn from a common external point to a circle are equal.
Subtended Angles and Arcs
- The subtended angle is the angle formed at a point by lines drawn from the ends of an arc (like focusing vision on an object).
Angles in the Same Segment
- Angles formed from the same arc and on the same side of the chord (in the same segment) are equal.
Arrowhead Theorem (Angle at Center)
- The angle at the center of the circle is twice the angle at the circumference, subtended by the same arc.
Cyclic Quadrilaterals
- A cyclic quadrilateral has all vertices on the circle.
- The opposite angles of a cyclic quadrilateral add up to 180°.
Alternate Segment Theorem
- The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.
Worked Example Summary
- To solve angle problems, use theorems such as the tangent-radius 90° rule, isosceles triangle properties, and the angle at center/circumference relationships.
Key Terms & Definitions
- Chord — Line joining two points on a circle.
- Tangent — Line touching a circle at only one point.
- Subtended Angle — Angle created at a point by lines from ends of an arc.
- Cyclic Quadrilateral — Four-sided shape with all vertices on the circle.
- Alternate Segment Theorem — Angle between tangent and chord equals angle in the alternate segment.
Action Items / Next Steps
- Practice solving circle theorem problems using these rules.
- Review and memorize key circle theorems for exams.