Chapter 10: Circles
Understanding Tangents to a Circle
A tangent is a line that touches a circle at exactly one point, unlike a secant which intersects the circle at two points. At the point of contact, the radius drawn to the tangent is always perpendicular to it. This means there’s only one unique tangent at any given point on a circle. When a wheel rolls on the ground, the point of contact between the wheel and the ground demonstrates this perpendicular relationship between the radius and the tangent.
Number and Properties of Tangents
The number of tangents that can be drawn depends on the location of the external point:
- No tangent if the point is inside the circle.
- Exactly one tangent if the point lies on the circle.
- Exactly two tangents if the point lies outside the circle.
Furthermore, the lengths of the two tangents drawn from an external point to a circle are always equal. This symmetry is used to solve problems involving chords, triangles, and constructions around circles.
Theorems and Applications
Several theorems build on the tangent concept:
- A tangent is perpendicular to the radius at the point of contact.
- The two tangents from an external point are equal in length.
- A line joining the center to the external point bisects the angle between the tangents.
These properties are applied in proving geometric relationships such as identifying rhombuses, bisecting chords, and constructing tangents. Multiple examples and exercises help reinforce these principles through problem-solving and logical reasoning.