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Theorems related to circle

Introduction to Circle Theorems

A circle is a set of points in a plane equidistant from a fixed point called the center. The study of circle geometry involves understanding relationships between elements such as chords, tangents, radii, and the angles they form. This chapter deals with the fundamental theorems related to these elements, which help in determining unknown lengths and angles and proving geometric properties.

The core theorems discussed are:

  • Angle subtended by a chord at the center and circumference 
  • Perpendicular from the center to a chord 
  • Equal chords and their distances from the center 
  • Tangent to a circle and its properties 

1. Angle Subtended by a Chord at the Center and at the Circumference

This theorem states that the angle subtended by a chord at the center of the circle is twice the angle subtended by the same chord at any point on the remaining part of the circle’s circumference.

  • Detailed Explanation:
     Let AB be a chord of the circle with center O. The angle formed at the center by the chord AB is ∠AOB, while the angle formed by the same chord at any point C on the circumference (not lying on the minor arc AB) is ∠ACB. According to the theorem,
     ∠AOB=2×∠ACB

 This relation holds true regardless of where point C lies on the circle’s circumference, as long as it is on the same segment defined by the chord.

 

  • Extended Example:
     Suppose chord AB subtends an angle of 120° at the center O of the circle. Then, any angle subtended by chord AB at the circumference on the same side as the center is 60°. If a problem requires finding the measure of an unknown angle at the circumference subtended by the same chord, this theorem provides a direct method to calculate it efficiently. 
  • Importance in Syllabus:
     This theorem is fundamental in solving various geometric problems involving circles, especially in determining unknown angles in cyclic quadrilaterals and circle segments. 
  • Keywords: angle subtended by chord, circle geometry, chord angle theorem, center angle, circumference angle 

2. Perpendicular from the Center to a Chord

The theorem regarding the perpendicular from the center to a chord states that the perpendicular drawn from the center of the circle to a chord bisects the chord.

  • Detailed Explanation:
     Consider a chord AB in a circle with center O. When a perpendicular OM is drawn from O to the chord AB, the point M is the midpoint of AB, i.e., AM = MB. This theorem reflects the symmetry in a circle, ensuring that the radius drawn perpendicularly to a chord divides the chord into two equal segments. 
  • Expanded Example:
     In a circle with center O, let chord AB be 12 cm long. Drawing a perpendicular OM from the center to the chord divides AB into two equal parts, each measuring 6 cm. If the length of OM is known or required, it can be used to find the radius using the Pythagoras theorem by considering triangle OMA. 
  • Syllabus Relevance:
     This theorem assists in constructing geometric figures accurately and proves vital in solving problems related to chord length and distance from the center. 
  • Keywords: perpendicular from center, chord bisector, circle chord property, radius perpendicular to chord 

3. Equal Chords and Their Distances from the Center

This theorem highlights the relationship between chord lengths and their distances from the center of the circle.

  • Theorem Statement:
     Chords that are equidistant from the center of a circle are equal in length, and conversely, equal chords are equidistant from the center. 
  • Detailed Explanation:
     If two chords AB and CD lie in a circle with center O such that the perpendicular distances OM and ON from O to chords AB and CD respectively are equal (OM = ON), then the lengths of the chords AB and CD are equal (AB = CD). Conversely, if two chords are equal in length, then their perpendicular distances from the center must also be equal. 
  • In-Depth Example:
     Consider two chords AB and CD in a circle with center O. If AB = CD = 10 cm, then the perpendicular distances OM and ON from O to these chords are also equal. This property helps in identifying congruent chords and proving geometric properties related to circle segments and symmetry. 
  • Use in Curriculum:
     This theorem is essential for establishing equal lengths and distances in circle geometry problems, supporting proofs and constructions. 
  • Keywords: equal chords, distance from center, chord length equality, circle geometry properties 

4. Tangent to a Circle and Its Properties

The tangent theorem is a cornerstone of circle geometry:

  • Theorem Statement:
     The tangent to a circle is perpendicular to the radius drawn to the point of contact. 
  • Comprehensive Explanation:
     If a tangent touches the circle at point P and OP is the radius drawn from the center O to P, then the tangent line at P is perpendicular to OP. This means,
     Tangent at P⊥OP
     This property forms the basis for solving problems related to tangent lengths, angle measures, and circle constructions. 
  • Expanded Example:
     In a circle with center O, a tangent touches the circle at point P. The radius OP measures 7 cm. The angle between the tangent line and radius OP is 90°, which can be used to solve problems involving tangents, such as finding lengths of tangents drawn from an external point using the Pythagoras theorem or other geometric relations. 
  • Curriculum Importance:
     This theorem supports the understanding of tangents and secants in circle geometry and is crucial for related problem-solving. 
  • Keywords: tangent to circle, radius and tangent, perpendicularity, circle tangent theorem 
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