Class 10 Notes - Circles
1. Introduction
This chapter builds on previous knowledge of circles and introduces the geometry of tangents. Understanding tangents is essential for applications in engineering, design, and geometry.
2. Key Terminology and Recap
- Circle: A set of all points at a fixed distance from a center.
- Radius (r): A line segment from the center to a point on the circle.
- Diameter (d): A line through the center touching two points on the circle (d = 2r).
- Chord: A line segment joining two points on the circle.
- Arc: A part of the circumference.
- Sector: Area enclosed by two radii and an arc.
- Segment: Area between a chord and its arc.
3. Tangent to a Circle
Definition: A tangent is a line that touches a circle at exactly one point.
- A tangent is perpendicular to the radius at the point of contact.
- Two tangents can be drawn from an external point and are equal in length.
- Tangents do not intersect the circle internally.
[Diagram: Circle with radius meeting tangent at right angle]
4. Number of Tangents from a Point
- Inside the circle: 0 tangents
- On the circle: 1 tangent
- Outside the circle: 2 tangents
5. Theorems Related to Circles
Theorem 1:
The tangent to a circle is perpendicular to the radius through the point of contact.
Proof Idea: Join center to point of contact, form right angle by contradiction or congruency.
Theorem 2:
From an external point, tangents drawn to a circle are equal in length.
Proof Idea: Use triangle congruency (RHS criterion) between two triangles formed by radii and tangents.
[Diagram: Circle with two tangents from an external point]
6. Construction Tips
Constructing Tangents from a Point Outside a Circle:
- Draw circle with center O and point P outside it.
- Join OP and find midpoint M.
- Draw a circle with center M and radius MO.
- Let this circle intersect the original at Q and R.
- Join PQ and PR. These are the tangents.
7. Real-Life Applications
- Gears and belt systems
- Circular tracks and road design
- Robotics and navigation paths
- Architectural domes
8. Problem Solving – Examples
Example 1:
Find the length of a tangent from a point 13 cm from the center of a circle of radius 5 cm.
Solution:
Use Pythagoras: l² = 13² - 5² = 169 - 25 = 144 ⇒ l = √144 = 12 cm
Example 2:
Prove triangle formed by tangent and radius is right-angled at the point of contact.
Solution: Since radius is perpendicular to tangent, angle at the contact point is 90°.
9. Tangents in Coordinate Geometry
For a circle centered at the origin (0, 0), the tangent line equations can be derived using algebra. These are more often covered in higher classes, but basic understanding is helpful.
10. Tangents to Multiple Circles
- Direct Common Tangents: Don’t cross between circles.
- Transverse Tangents: Pass between circles.
- Number of tangents depends on distance between centers.
11. CBSE Exam Focus
Questions often include:
- Proof-based theorems
- Length of tangents using Pythagoras theorem
- Geometric construction questions
12. Practice Questions
Short Answer:
- Define tangent. State one property.
- How many tangents from a point inside the circle?
- Label parts of a circle on a diagram (radius, tangent, chord, etc.)
Long Answer:
- Prove tangents from external point are equal.
- Find length of tangents from a point 10 cm from the center of a circle with radius 6 cm.
- Construct tangents to a circle from an external point 8 cm away.
13. HOTS (Higher Order Thinking Skills)
- Point A lies 5 cm from center. Radius = 3 cm. Find area of triangle OAP with PA as tangent.
- Prove that angle between tangents from external point T to points P and Q is bisected by line TO (center).
14. Summary
- Tangents touch the circle at one point.
- Tangent is perpendicular to the radius.
- Two equal tangents can be drawn from an external point.
- Proof and construction form the key focus of this chapter.