ICSE Class 10 Maths Chapter 19 Chord Properties of A Circles

Chapter 19

Chord Properties Of A Circles

Points To Remember

Circle. A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point remains constant.

The fixed point is called the centre and the constant distance is called the radius of the circle.

The given figure consists of a circle with centre O and radius equal to r units.

Circumference. The perimeter of a circle is called its circumference.

Circumference = 2\(\pi\) r

Radius. A line segment joining the centre and a point on the circle is called its radius.

The plural of radius is radii.

In the given figure, OA, OB and OC are the radii of a circle.

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

Diameter. A chord of the circle passing through the centre of a circle is called its diameter.

In the adjoining figure, AOB is a diameter of a circle with centre O.

Diameter is the largest chord of a circle.

All diameters of a circle are equal in length.

Diameter = 2 \(\times\) Radius.

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

In the given figure, the line l cuts the circle in two points C and D. Then, l is a secant of the circle.

Teacher's Note

Understanding circles and their properties helps us recognize circular patterns in everyday life, from the wheels on vehicles to the round plates we use at home.

Tangent. A line that intersects the circle in exactly one point is called a tangent to the circle.

The point at which the tangent intersects the circle is called its point of contact. In the given figure, SPT is a tangent at the point P of the circle with centre O. Clearly, P is the point of contact of the tangent with the circle.

Facts about Tangents :

(i) No tangent can be drawn to a circle through a point inside it.

(ii) One and only one tangent can be drawn to a circle at a point on it.

(iii) Two tangents can be drawn to a circle from a point outside it.

In the given figure, PT and PS are the tangents to the circle from point P.

Position of a Point With Respect to a Circle

Let us consider a circle with centre O and radius r.

A point P is said to lie:

(i) inside the circle, if OP < r ;

(ii) on the circle, if OP = r.

(iii) outside the circle, if OP > r.

In the adjoining figure of a circle with centre O and radius r.

(i) The points A, O, B lie inside the circle ;

(ii) The points P, Q, R lie on the circle ;

(iii) The points X, Y, Z lie outside the circle.

Interior and Exterior of a Circle :

The region consisting of all those points which lie inside a circle, is called the interior of the circle. The region consisting of all those points which lie outside a circle, is called the exterior of the circle.

Circular Region or Circular Disc :

The region consisting of all those points which are either on the circle or lie inside the circle, is called the circular region.

Arc. A continuous piece of a circle is called an arc of the circle.

Let P and Q be any two points on a circle with centre O.

Then, clearly the whole circle has been divided into two pieces, namely arc PAQ and arc QBP, to be denoted by \(\overarc{PAQ}\) and \(\overarc{QBP}\) respectively.

We may denote them by \(\overarc{PQ}\) and \(\overarc{QP}\) respectively.

An arc PQ is called a minor arc or a major arc, according as length.

\((\overarc{PQ}) < \text{length} (\overarc{QP})\) or length \((\overarc{PQ}) >\) length \((\overarc{QP})\)

Teacher's Note

Recognizing arcs in circles helps us understand how to measure portions of circular paths, which is useful when calculating distances on circular tracks or roads.

Central Angle. An angle subtended by an arc at the centre of a circle is called its central angle.

In the given figure, central angle of \(\overarc{PQ} = \angle POQ = \theta\)

Degree Measure of an Arc

Let \(\overarc{PQ}\) be an arc of a circle with centre O.

If \(\angle POQ = \theta°\), we say that the degree measure of \(\overarc{PQ}\) is \(\theta°\) and we write, \((\overarc{PQ}) = \theta°\).

If m \((\overarc{PQ}) = \theta°\), then m \(\overarc{QP} = (360 - \theta)°\).

Degree measure of a circle is 360°.

Congruent Arcs. Two arcs AB and CD are said to be congruent, if they have same degree measure.

\(\overarc{AB} \cong \overarc{CD} \Leftrightarrow \text{m} (\overarc{AB}) = \text{m} (\overarc{CD}) \Leftrightarrow \angle AOB = \angle COD.\)

Congruent Circles. Two circles of equal radii are said to be congruent.

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

Semi-Circle. A diameter divides a circle into two equal arcs. Each of these two arcs is called a semi-circle.

The degree measure of a semi-circle is 180°.

An arc whose length is less than the arc of a semi-circle is called a minor arc, otherwise it is called a major arc.

In the given figure of a circle with centre O, \(\overarc{ABC}\) as well as \(\overarc{ADC}\) is a semi-circle.

Segment. A segment is a part of a circular region bounded by an arc and a chord, including the arc and the chord.

The segment containing the minor arc is called a minor segment, while the other one is major segment.

The centre of the circle lies in the major segment.

Alternate Segments of a Circle. The minor and major segments of a circle are called alternate segments of each other.

Sector of a Circle : The part of the plane region enclosed by an arc of a circle and its two bounding radii is called a sector of the circle.

Thus, the region OABO is the sector of a circle with centre O.

Quadrant. One-fourth of a circular disc is called a quadrant.

Cyclic Quadrilateral. If all the four vertices of a quadrilateral lie on a circle, then such quadrilateral is called a cyclic quadrilateral.

If four points lie on a circle, they are said to be concyclic.

We can also say that quad. ABCD is inscribed in a circle with centre O.

Teacher's Note

Cyclic quadrilaterals appear in many real-world structures, such as the corners of a square garden or the vertices of a diamond-shaped kite, helping us understand geometric properties in practical contexts.

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