ICSE Class 8 Maths Geometry Chapter 07 Circle

Circle

A circle is a closed plane curve consisting of all points at the same (or constant) distance from a fixed point O of the plane. The fixed point O is called the centre of the circle and the constant distance of the points of the circle from its centre is called the radius of the circle. In the figure, O is the centre of the circle and A, B, C and D are points on the circle.

OA = OB = OC = OD = the radius of the circle = r (say)

A line segment joining the centre and a point on the circle is also known as a radius. Thus, OA, OB, OC and OD are radii (plural of radius) of the circle.

The symbol for a circle is ⊙. Any three points on a circle can be used to name it. Thus, the circle in the figure may be called ⊙ABC or ⊙ABD or ⊙ACD or ⊙BCD. A circle with centre O and radius r is denoted by C(O, r).

Exterior And Interior Of A Circle

In the figure, r is the radius of the circle which has its centre at O. Consider a point P in the plane of the circle.

If OP < r then P lies inside the circle.

The set of all such points in the plane of the circle which are at a distance less than r from the centre of the circle is called the interior of the circle.

Let Q be a point in the plane of the circle.

If OQ > r then Q lies outside the circle.

The set of all such points in the plane of the circle which are at a distance greater than r from the centre of the circle is called the exterior of the circle.

The interior of a circle together with the points on the circle is called a circular region.

Chord Of A Circle

A line segment joining two points on a circle is called a chord of the circle. In the figure, PQ, PR and PS are chords of the circle.

Diameter Of A Circle

A chord that passes through the centre of a circle, is called a diameter of the circle. In the figure above, PQ is a chord passing through the centre O. So, the line segment PQ is a diameter of the circle. The length of the line segment PQ is also called the diameter of the circle. In other words, the diameter is the length of the longest chord of a circle.

Now, PQ = PO + OQ = radius + radius = 2 - radius. Therefore,

radius = diameter / 2

diameter = 2 - radius

Arc Of A Circle

A continuous part of a circle is called an arc of the circle. In the figure, PQ is an arc. The symbol for an arc is ⌢. Thus, PQ denotes the arc PQ.

Circumference Of A Circle

The length of the boundary of a circle is called the circumference of the circle.

Semicircle

A semicircle is half a circle formed when a circle is cut along a diameter. In the figure, arcs PRQ and PSQ are two semicircles.

Minor And Major Arcs Of A Circle

When P and Q are two points on a circle not at the ends of a diameter, the smaller part of the circle from P to Q is called the minor arc and the larger part from P to Q is called the major arc.

Angle Subtended By An Arc

In the figure, A is a point on a circle which has its centre at O. PQ is an arc of the circle. Then ∠PAQ is the angle subtended by PQ at the point A of the circle. The ∠POQ is the angle subtended by PQ at the centre O of the circle.

Central Angle

The angle subtended by any arc of a circle at the centre of the circle is called a central angle on the arc. In the figure, ∠POQ, ∠QOR and ∠POR are all central angles.

Secant And Tangent

A straight line which intersects a circle at two distinct points is called a secant of the circle. In the figure, PQ is a secant of the circle with centre at O.

A straight line that intersects a circle at only one point is called a tangent to the circle at that point. In the figure, the line BA is a tangent to the circle at A. The length of the line segment BA is known as the length of the tangent from B and the point A is known as the point of contact of the tangent.

Property 1: The tangent at any point of a circle and the radius through the point are perpendicular to each other.

In the figure, ∠OAB = 90°.

Property 2: There is one, and only one, tangent at any point of a circle.

In the figure, since BA and CA are tangents to the circle at A, CAB is a straight line.

Property 3: Two tangents can be drawn to a circle from a point outside the circle.

In the figure, AD and AE are two tangents to the circle from the point A outside the circle.

Property 4: If two tangents are drawn to a circle from a point outside the circle then the tangents are equal in length.

In the figure, AD and AE are two tangents to the circle drawn from A outside the circle, so AD = AE.

Sector Of A Circle

A sector of a circle is the region bounded by two radii of the circle and either of the arcs that they cut off. In the figure, OP and OQ are two radii. They cut off the minor arc PSQ and the major arc PRQ. The region bounded by OP, OQ and PSQ is a minor sector. The region bounded by OP, OQ and PRQ is a major sector.

Segment Of A Circle

The region bounded by a chord of the circle and one of the arcs cut off by the chord is called a segment of the circle. In the figure, the chord PQ cuts the circle into the minor arc PRQ and the major arc PSQ. The smaller part PRQ P of the circular region is called the minor segment while the bigger part PSQP is called the major segment.

Angle In A Semicircle

In the figure, PQ is a diameter of a circle which has its centre at O. R and S are two points on the circle. ∠PRQ and ∠PSQ are examples of angles in a semicircle.

Property: An angle in a semicircle is a right angle.

In the figure, ∠PRQ = ∠PSQ = 90°.

Example

In the adjoining figure, find AB and the radius of the circle.

Solution: An angle in a semicircle = 90° - ∠ACB = 90°. In the right-angled triangle ABC,

AB² = AC² + BC² = (12² + 16²) cm² = 400 cm²

AB = 20 cm. So, the radius = (1/2) - AB = (1/2) - 20 cm = 10 cm.

Solved Examples

Example 1: In the adjoining figure, find OM.

Solution: MN is a tangent to the circle at M. So, OM ⊥ MN. Now, in the right ∆OMN, OM² + MN² = ON²

OM = √(ON² - MN²) = √(17² - 8² ) cm = √(289 - 64) cm = √225 cm = 15 cm.

Example 2: In the figure, OA = 6.5 cm and AC = 5 cm. Find BC.

Solution: An angle in a semicircle = 90° - ∠ACB = 90°. AB = 2 - OA = 2 - 6.5 cm = 13 cm. In the right-angled triangle ABC, AB² = AC² + BC²

BC = √(AB² - AC²) = √(13² - 5² ) cm = √(169 - 25) cm = √144 cm = 12 cm.

Exercise 7A

1. Identify the false statements.

(i) The radius of a circle is half of its diameter.

(ii) The diameter of a circle is the length of its longest chord.

(iii) A secant of a circle intersects a circle at three distinct points.

(iv) A tangent to a circle can touch the circle at more than one point.

(v) If O is the centre of the circle with diameter 4 cm and OP = 3 cm then the point P lies in the interior of the circle.

(vi) An arc of a circle may be a discontinuous part of the circle.

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