Circles

Circles

A circle is a collection of all points in a plane which are at a constant distance from a fixed point.

1. Centre: The fixed point is called the centre of the circle.

2. Radius: A line segment joining the centre of the circle to a point on the circle is called its radius. The circle has infinite number of radii. All radii of a circle are equal.

3. Chord: A line segment joining any two points on the circle is called a chord.

4. Diameter: A chord passing through the centre of circle is called its diameter. Diameter is the longest chord of the circle.

d = 2r

The diameter of a circle = twice the radius of the circle

5. Arc: A part of a circle is called an arc.

  • Minor arc: An arc of a circle whose length is less than that of a semicircle of the same circle.
  • Major arc: An arc of a circle whose length is greater than that of a semicircle of the same circle is called a major arc.

6. Semicircle: Diameter of a circle divides a circle into two equal arcs, each of which is called a semicircle.

7. Sector: The region bounded by an arc of a circle and two radii at its end points is called a sector.

8. Segment: A chord divides the interior of a circle into two parts. Each of which is called a segment.

9. Circumference: The length of the boundary of a circle is the circumference of the circle.

The ratio of the circumference of circle to its diameter is always a constant, which is denoted by Greek letter π.

$$ \frac{c}{d} = \frac{c}{2r} = \pi $$

10. Concentric Circles: Circles having the same centre but different radii are called concentric circles.

Important Rules

  1. Two arcs of a circle are congruent if and only if the angles subtended by them at the centre are equal.
  2. Two arcs of a circle are congruent if and only if their corresponding chords are equal.
  3. Equal chords of a circle subtend equal angles at the centre and conversely if the angles subtended by the chords at the centre of a circle are equal, then the chords are equal.
  4. The perpendicular drawn from the centre of a circle to a chord bisects the chord. Conversely the line joining the centre of a circle to the mid-point of a chord is perpendicular to the chord.
  5. Equal chords of a circle are equidistant from the centre, conversely chords that are equidistant from the centre of a circle are equal.
  6. There is one and only one circle passing through three non-collinear points.

Angles in a Circle

1. Central Angle: The angle made at the centre of a circle by the radii at the end points of an arc (or a chord) is called the central angle or angle subtended by an arc (or chord) at the centre.

The length of an arc is closely associated with the central angle subtended by the arc. The degree measure of a minor arc of a circle is the measure of its corresponding central angle.

The degree measure of a semicircle is 180° and that of a major arc is 360° minus the degree measure of the corresponding minor arc.

Length of an arc = circumference × (degree measure of the arc)/360°

2. Inscribed Angle: The angle subtended by an arc or chord on any point on the remaining part of circle is called an inscribed angle.

Important Properties

  1. The angle subtended at the centre of a circle by an arc is double the angle subtended by it on any point on the remaining part of the circle.
  2. Angle in a semicircle is a right angle.
  3. Angles in the same segment of a circle are equal.

Concyclic Points

Points which lie on a circle are called concyclic points.

If you take a point P, you can draw not only one but many circles passing through it. You can also draw as many circles as you wish, passing through the two points.

Now take three points P, Q and R which do not lie on the same straight line. In this case you can draw only one circle passing through these three non-colinear points.

Now take four points P, Q, R, and S which do not lie on the same line. You will see that it is not always possible to draw a circle passing through four non-collinear points.

  1. Given one or two points there are infinitely many circles passing through them.
  2. Three non-collinear points are always concyclic and there is only one circle passing through all of them.
  3. Three collinear points are not concyclic (or noncyclic).
  4. Four non-collinear points may or may not be concyclic.

Cyclic Quadrilateral

A quadrilateral is said to be a cyclic quadrilateral if there is a circle passing through all its four vertices.

If a pair of opposite angles of a quadrilateral is supplementary, then the quadrilateral is cyclic.

Sum of the opposite angles of a cyclic quadrilateral is 180°.

If PQRS is a cyclic quadrilateral, ∠P + ∠R = 180° or ∠Q + ∠S = 180°.

If PQRS is a cyclic parallelogram, then it is a rectangle.

Secants and Tangents

When a line and a circle co-exist in the same plane, there can be three distinct possibilities:

  1. The line and the circle have no common point. The line does not intersect the circle at all. The line lies in the exterior of the circle.
  2. The line intersects the circle in two distinct points.
  3. The line intersects the circle in only one point.

Secant: A line which intersects the circle in two distinct points is called a secant.

Tangent: A line which touches a circle at exactly one point is called a tangent and the point where it touches the circle is called the point of contact.

Tangent as Limiting Case

  1. When two points of intersection of secant and circle coincide it becomes a tangent.

Tangent and Radius through Point of Contact

  1. A radius, though the point of contact of tangent to a circle, is perpendicular to the tangent at that point.

Tangents from Point Outside the Circle

  1. From an external point, two tangents can be drawn to a circle.
  2. The lengths of two tangents from an external point are equal.
  3. The tangents drawn from an external point to a circle are equally inclined to the line joining the point to the centre of circle.

Intersecting Chords

If two chords AB and CD of a circle intersect at a point P outside or inside the circle, then

PA × PB = PC × PD

Intersecting Secant and Tangent

If PAB is a secant to a circle intersecting the circle at A and B and PT is a tangent to the circle at T, then

PA × PB = PT2

Angles by Tangent and Chord

The angles made by a chord in alternate segment through the point of contact of a tangent is equal to the angle between chord and tangent.

If a line makes with a chord angles which are equal respectively to the angles formed by the chord in alternate segments, then the line is a tangent to the circle.

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