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CIRCLES
Circle: The collection of all points in a plane, which are at fixed distance from a fixed point in the plane is called a circle. The fixed point is called the center of the circle and the fixed distance from the center is called the radius of the circle. In the given figure, O is the center and OA is the radius of the circle.
TERMS RELATED TO THE CIRCLE
Chord: A line segment joining two points on the circle is called the chord of the circle. In figure (i), PQ is the chord of the circle.
Diameter: It is the longest chord of the circle, it passes through the center of the circle and two times the radius. AOB or AB is a diameter in Figure (i).
Arc: A part of a circle between two points on it is called an arc. Smaller part of the circle is called a minor arc, whereas the longer part of the circle if called a major arc. In figure (ii), PQ is the minor arc and PRQ is the major arc.
In figure (ii), PQ is the minor arc and PRQ is the major arc.
Sector: The region between arc and two radii is called sector as shown in figure (iii).
Minor arc: Minor arc corresponds to minor sector and major arc corresponds to major sector as shown in the figure (iv).
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Segments: the region between a chord and either of its arcs is called the segment of the circle.
A circle has two segments:
Major segment and Minor Segment as shown in figure (iii).
Parts of the circle
A circle divides the plane on which it lies into three parts.
(i) Inside the circle and the interior.
(ii) Outside the circle called exterior.
(iii) The circle.
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ANGLES SUBTENDED BY A CHORD AT A POINT
Circles having the same center are called concentric circles.
Two circles are congruent if they have equal radii.
Theorem 1: Equal chords of a circle subtend equal angles at the center.
Given: AB=CD and O is the center of the circle.
To prove: ∠AOB= ∠COD
Proof:
∠AOB= ∠COD (Given)
OA= OD (Radii of same circle)
OB=OC (Radii of same circle)
AB=CD (CPCT)
PERPENDICULAR FROM THE CENTER TO A CHORD
Theorem 3: The perpendicular drawn from the center of a circle to a chord bisects the chord.
Theorem 4: The line drawn through the center of a circle to bisect a chord is perpendicular to the chord. In the adjoining figure:
‘O’ is the center of the circle and AC=BC.
Circle through three points:
There is one and only one circle passing through three given non-collinear points.
If ABC is a triangle, there is a unique circle passing through three vertices A, B and C of the triangle. This circle is called the circumcircle of the
Its center and the radius are called the circumcenter and the circumradius of the triangle .
In the adjoining figure, ‘O’ is the circumcenter and OA, OB and OC are the circumradius of the circumcircle of
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EQUAL CHORDS AND THEIR DISTANCES FROM THE CENTER
The length of the perpendicular from a point to a line is the distance of the line from the point.
Equal chords of a circle (or of congruent circles) are equidistant from the center (or centers). Conversely, chords equidistant from the center of the circle are equal in the length.
ANGLES SUBTENDED BY AN ARC OF A CIRCLE AND CYCLIC QUADRILATERAL
- If two chords of a circle are equal, then their corresponding arcs are congruent and conversely, if two arcs are congruent, then their corresponding chords are equal.
- Congruent arcs (or equal arcs) of a circle subtend equal angles at the center.
- The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
- Angles in the same segment of a circle are equal. Conversely, if a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle (i.e. they are concyclic).
- Angle in a semicircle is a right angle.
- A quadrilateral is called cyclic if all the four vertices of it lie on the circle.
- The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degree . Conversely, if the sum of pair of opposite angles of a quadrilateral is 180degree , the quadrilateral is cyclic.
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Table of Contents
CIRCLES CLASS 9 QUESTIONS FOR CBSE STUDENTS
1. The centre of a circle lies in ________of the circle.
Ans. Interior
2. A point, whose distance from the centre of a circle is greater than its radius lies in ____________ of the circle.
Ans. Exterior
3. The longest chord of a circle is a __________ of the circle.
Ans. Diameter
4. An arc is a __________ when its ends are the ends of a diameter.
Ans. Semicircle
5. Segment of a circle is the region between an arc and ___________ of the circle.
Ans. Chord
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6. A circle divides the plane, on which it lies, in _________parts.
Ans. Three
7. Line segment joining the centre to any point on the circle is a radius of the circle.
Ans. True
All the line segment from the centre to the circle is of equal length and is called radius of the circle.
8. A circle has only finite number of equal chords.
Ans. False
We can draw infinite number of equal chords.
9. If a circle is divided into three equal arcs, each is a major arc.
Ans. False
All three arcs will be minor arcs.
10. A Chord of a circle, which is twice as long as its radius, is a diameter of the circle.
Ans. True
A chord which is twice as long as radius must pass through the centre of the circle and is diameter to the circle.
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