CIRCLES

TOPIC: CIRCLES

Learning Objectives.

1. recognize various parts of a circle.
2. state the properties of chords of a circle.
3. state and apply the property of angles at the centre.
4. state and apply the property of angles in the same segment.
5. recognize the property of angles in a semi-circle.
6. explain the meaning of the concyclic points.
7. state the properties of angles in a cyclic quadrilateral.
8. state the properties of angles in a cyclic quadrilateral.
9. state the definition of a tangent to a circle.
10. recognize the properties of the tangents to a circle.
11. state and apply the alternate segment theorem.

 

Circles

            A circle is a set of ponts in a plane so that they are always at a constant distance rom a fixed point. The fixed point is known as the centre and the distance is known as radius (plural: radii).

Parts of a circle

A circle is a closed curve in a plane such that all points on the curve are equidistant from a fixed point. The given distance is called the radius of the circle.

A chord is a line segment with its end points on the circle and a diameter is a chord passing through the centre.

An arc is a part of the circle. A segment is the region bounded by a chord and an arc of the circle.

A sector is the region bounded by two radii and an arc. It is a part of a circular region bounded by the two radii and the arc subtended by them.

Central Angles

The angle formed by two radii of a circle at the centre is called a central angle. 

In the figure below, <POQ is a central angle. Arc PQ subtends <POQ at the centre. An arc whose measure is 45° subtends an angle 45° of at the centre. Also, <POQ intercepts the arc PQ at the circumference. 

 

The length of an arc is proportional to the measure of the central angle, provided the radius of the circle is not changed. 

Chords of a circle

 A chord is a line segment whose end-points are on the circle. A chord which passes through the centre of the circle is called a diameter.