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Theorems Related to Tangent to a Circle

Description:

Chapter 15, “Theorems Related to Tangent to a Circle,” is a core part of Class 10 Mathematics under the West Bengal Board (WBBSE). This chapter explains the fundamental geometrical properties of tangents drawn to a circle. It introduces key theorems, their geometric proofs, and their applications in solving numerical and theoretical problems. Students learn to work with lines that just touch a circle (tangents), understand the concept of perpendicularity with the radius, and explore equal lengths from external points.

1. Understanding the Tangent to a Circle

1.1 Definition:

A tangent to a circle is a straight line that touches the circle at exactly one point. This point is called the point of contact.

1.2 Key Properties:

  • A tangent never enters the interior of the circle. 
  • It touches the circle at one unique point. 
  • A circle can have an infinite number of tangents from various points outside its circumference. 

2. Theorem 1: The Tangent is Perpendicular to the Radius

2.1 Statement:

“The tangent to a circle is perpendicular to the radius drawn to the point of contact.”

2.2 Diagram Description (for learning):

  • Let a circle with centre O be given. 
  • Let a tangent line touch the circle at point P. 
  • The radius OP is drawn. 

Then, OP ⊥ Tangent at P.

2.3 Proof Outline:

By using the concept of shortest distance, and Euclidean geometry postulates, it can be proved that the line from the centre to the point of contact is perpendicular to the tangent line.

2.4 Applications:

  • Used to construct tangents from a point. 
  • Helps in finding the angle between the radius and a line. 
  • Common in geometry problems involving circles. 

3. Theorem 2: Lengths of Tangents from an External Point are Equal

3.1 Statement:

“The lengths of two tangents drawn from an external point to a circle are equal.”

3.2 Diagram Description:

  • Let PA and PB be two tangents drawn from an external point P to a circle with centre O. 
  • Points A and B are the points of contact. 

Then, PA = PB.

3.3 Proof Outline:

Using triangle congruence (ΔOPA ≅ ΔOPB), and by proving sides equal using SSS or RHS congruence, the equality of the two tangents is established.

3.4 Applications:

  • Used in geometrical construction. 
  • Helps in calculating unknown lengths in circular figures. 
  • Important in proving congruence or similarity in complex diagrams. 

4. Construction and Application-Based Concepts

4.1 Drawing Tangents from an External Point:

Steps:

  1. Draw the circle with centre O. 
  2. Mark an external point P. 
  3. Draw the line OP. 
  4. Bisect OP; mark midpoint M. 
  5. With M as centre and MO as radius, draw a circle. 
  6. This circle intersects the original circle at A and B. 
  7. Draw lines PA and PB — these are the required tangents. 

Outcome: PA and PB are equal-length tangents from point P to the circle.

4.2 Use in Coordinate Geometry:

Although not deeply covered in Class 10, the concept helps in understanding advanced geometrical contexts like equations of tangents, circle equations, etc.

5. Properties of Tangents – At a Glance

Property Explanation
Tangent touches at one point only                               Called the point of contact
Perpendicularity                                       90° angle made
Equal tangents                                               PA = PB
Infinite tangents A circle has infinitely many tangents, but from a single external point, only two distinct tangents can be drawn

6. Real-Life Relevance

While theoretical, these properties are used in:

  • Engineering and architecture for designing round structures 
  • Navigation systems that involve circular paths 
  • Wheel and axle mechanism analysis 
  • Road design involving curves and touchpoints 

 

7. Practice Example Questions

Example 1:

Prove that the tangents drawn from an external point to a circle are equal in length.

Solution Sketch:  Draw two tangents from a point, join the centre to points of contact, show triangle congruence → SSS → lengths are equal.

Example 2:

In a circle with centre O, a tangent touches the circle at point A. Show that ∠OAP = 90°.

Solution:  Use the perpendicularity theorem. Radius OA is ⊥ to tangent at point A.

8. Summary

Chapter 15, “Theorems Related to Tangent to a Circle,” teaches two foundational geometric theorems:

  1. Tangent is perpendicular to radius 

Tangents from external points are equal in length

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