15. Circle, Circular Region, Segment, arc, Properties of a Circle

 

CIRCLE, CIRCULAR REGION, SEGMENT, ARC, PROPERTIES OF A CIRCLE

Learn and Remember :

    Circle : The set of points, in a plane, equidistant from a fixed point in that plane is called a circle.

The fixed point is called the center of the circle.

1. Radius : A line segment joining the center of the circle and any point on the circle is called the radius of that circle.

For example, seg OK is a radius.

2. Chord : A segment joining any two points on a circle is called its chord. For example, seg MN is a chord of a circle.

3. Diameter :

(i) A chord passing through the center of a circle is called its diameter.

(ii) The length of a diameter is twice that of its radius.

(iii) A diameter is the longest chord of a circle.

4. Circular region : The points on the circle and the points in the interior of the circle together make the circular region. In the figure the shaded region is the circular region.

5. Segment of the circle : A chord divides the circular region into two parts. Each part is called a segment. In the figure, the shaded portion AXB is the minor segment. The unshaded portion AYB is the major segment.

6. Arc : The circle is divided into two parts by taking any two points on the circle. Each part is called an arc. In the figure, points A and B are common points to both the arcs. In order to avoid confusion, which of the two arcs are we considering, a third point is taken to name the arc.

In the figure, arc ACB and arc ADB are shown.

7. Semicircle : The diameter of a circle divides the circle into two arcs of equal measure. Each of the arcs is called a semicircle. In the figure, arc PYQ and arc PXQ are the semicircles.

8.Central angle : An angle having the center of the circle as its vertex is called a central angle.

In the figure, ∠AOB is the central angle. m ∠AOB = m(arc AXB).

9. Measure of an arc :

(i) The measure of a minor arc : The measure of the minor arc is equal to the measure of the central angle.

m(arc PXR) =m ∠POR.

The measure of a minor arc is less than 180°.

(ii) The measure of a major arc : The measure of a major arc = 360° – the measure of the minor arc.

The measure of a major arc is greater than 180° but less than 360°.

(iii) The measure of a semicircular arc : The measure of a semicircular arc is 180°.

10. Inscribed angle : In the circle with center O, the vertex B of  ∠ABC is a point of arc ABC and the end points A and C of the arc are on the side of the angle. ∠ABC is inscribed in the arc ABC. Arc ADC is the intercepted arc by ∠ABC.

m ∠ABC =  ½ m(arc ADC).

11. An angle subtended by a semicircle at a point on the circle is a right angle. In the figure, ∠BAC = 90°.

12. Cyclic quadrilateral : A quadrilateral whose vertices lie on a circle is called a cyclic quadrilateral. In the figure, ☐ABCD is a cyclic quadrilateral. The opposite angles of a cyclic quadrilateral are supplementary.

m ∠A+ m ∠C =180°; m B + m ∠D= 180°.

13. The circumference of a circle : The length of the boundary of a circle (i.e. the perimeter) is called the circumference of a circle.

The circumference of a circle = 2ℼr or ℼd.

14. The area of a circle : It means the area of the circular region. The area of a circle = ℼr×r (r square )

15. The distance between the center of a circle and a chord is the length of perpendicular drawn from the center to the chord. In the figure, the distance of chord AB from the center o of the circle is OP.

16. The properties of a circle and a chord :

    (i) The perpendicular drawn from the center of a circle to its chord bisects the chord.

In the figure, seg OP ⊥ chord AB.

:: I(AP) =l(BP).

 (ii) In a circle, congruent chords are equidistant from the center of the circle.

In the figure, chord AB ≅ chord CD.

 l(OP) ≅ ((OQ).

(iii) Congruent chords of a circle form congruent angles at the centre of the circle. In the figure, chord AB ≅ chord CD.

:: LAOB ≅ ∠COD.

17. Angles in the same segment :

Angles in the same segment are congruent. For example, ∠CAD and ∠CBD are angles in the same segment. Therefore, m ∠CAD = m ∠CBD.

18. Angles in a semicircular region :

The angle in a semicircular region is a right angle. e.g. AYB is a semicircular region and m ∠APB = 90°.

19.Angles in the minor and major segments : The angle in the minor segment is an obtuse angle. The angle in the major segment is an acute angle. In the figure, ∠ADB is in the minor segment AYB. Hence, ∠ADB is an obtuse angle. ∠ACB is in the major segment AXB. Therefore, ∠ACB is an acute angle.