Circles: Important Concepts for CBSE Class 10
The chapter 'Circles' in Class 10 Mathematics primarily focuses on the properties of tangents to a circle. Understanding these concepts is fundamental for solving geometry problems and excelling in your CBSE Board Exam.
1. Basic Definitions Related to Circles
- Circle: A collection of all points in a plane which are at a fixed distance from a fixed point in the plane.
- Radius: The fixed distance from the center to any point on the circle.
- Chord: A line segment joining any two points on a circle.
- Diameter: A chord passing through the center of the circle (longest chord).
- Arc: A continuous piece of a circle.
- Segment: The region between a chord and its corresponding arc.
- Sector: The region between two radii and their corresponding arc.
- Secant: A line that intersects a circle at two distinct points.
- Tangent: A line that intersects the circle at exactly one point (the point of contact).
2. Theorems Related to Tangents
- Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
If $OP$ is the radius and $AB$ is the tangent at point $P$, then $OP \perp AB$. - Theorem 2: The lengths of tangents drawn from an external point to a circle are equal.
If $PA$ and $PB$ are two tangents drawn from an external point $P$ to a circle with center $O$, then $PA = PB$.
3. Number of Tangents from a Point on a Circle
- Point inside the circle: No tangent can be drawn.
- Point on the circle: Exactly one tangent can be drawn.
- Point outside the circle: Exactly two tangents can be drawn.
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