Revision Notes of Chapter 10 Circles Class 9th Math
Topics in the Chapter- Terms related to Circles
- Fundamentals of Circles
- Theorems
- The locus of a point which moves in a plane in such a manner that its distance from a given fixed point is always constant, is called a circle.
- The fixed point is called the centre and constant distance is called the radius of the circle.
In the figure, ‘O’ is centre and OP = r is a radius. We denote it by C(O, r).
- A line segment, terminating (or having its end points) on the circle is called a chord. A chord, passing through the centre is called a diameter of the circle.
- A line which intersects a circle in two distinct points is called a secant of the circle.
- A line intersecting the circle in exactly one point is called a tangent to the circle.
- In the figure, PQ is a chord, AB is a diameter, XY is a secant and ST is a tangent to the circle at C.
(i) Diameter is the longest chord in a circle.
(ii) Diameter = 2 × Radius
- The length of the complete circle is called its circumference, whereas a piece of a circle between two points is called an arc.
Note:
(i) A diameter of a circle divides it into two equal arcs, each of which is called a semicircle.
(ii) If the length of an arc is less than the semicircle, then it is a minor arc, otherwise, it is a major arc.
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(i) A diameter of a circle divides it into two equal arcs, each of which is called a semicircle.
(ii) If the length of an arc is less than the semicircle, then it is a minor arc, otherwise, it is a major arc.
- The region consisting of all points lying on the circumference of a circle and inside it is called the interior of the circle.
- The region consisting of all points lying outside a circle is called the exterior of the circle.
- The region consisting of all points which are either on the circle or lie inside the circle is called the circular region.
- A chord of a circle divides it into two parts. Each part is called a segment.
- The part containing the minor arc is called the minor segment, and the part containing the major arc is called the major segment.
- A quadrilateral of which all the four vertices lie on a circle is called a cyclic quadrilateral. The four vertices A, B, C and D are said to be concyclic points.
Fundamentals of Circles
- Equal chords of a circle (or of congruent circles) subtend equal angles at the centre.
- If two chords of a circle subtend equal angles at the centre, then the chords are equal.
- The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
- The perpendicular from the centre of a circle to a chord bisects the chord.
- Equal chords of a circle are equidistant from the centre whereas the equidistant chords from the centre are equal.
- Chords corresponding to equal arcs are equal.
- Congruent arcs of a circle subtend equal angles at the centre.
- The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part.
- Angles in the same segment are equal, whereas the angle in a semicircle is a right angle.
- The sum of either pair of opposite angles of a cyclic quadrilateral is 180º.
- If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.
- 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, then the four points are cyclic.