This chapter explores the fascinating world of circles, their properties, and their applications. You'll learn about the basic definitions and symmetries of circles, how to construct circles through given points, and the properties of chords, arcs, and angles in circles. The chapter also covers cyclic quadrilaterals, tangents, and secants, as well as the calculation of circumference, area, and arc length. Understanding these concepts will help you appreciate the beauty and utility of circles in mathematics and the real world.
What you'll learn
1Understand the definition and basic properties of a circle
2Identify and describe the symmetries of a circle
3Determine the number of circles that can pass through given points
4Calculate the length of chords and the angles they subtend
5Prove and apply theorems related to chords and their distances from the centre
6Understand the concept of concyclic points and cyclic quadrilaterals
7Calculate the angles subtended by arcs at the centre and at points on the circle
Chapter at a glance
01Understanding Circles and Circular Motion
02Chords, Arcs, and Angles in Circles
03Cyclic Quadrilaterals and Their Properties
04Cyclic Quadrilaterals and Their Properties
05Tangents to Circles and Secants
06Circumference, Area, and Arc Length
Detailed chapter notes
01
Understanding Circles and Circular Motion
A circle is a set of all points in a plane that are at a given distance, called the radius, from a fixed point called the centre. Every circle has a centre, and all points on the circle are at equal distance from the centre. Circles are perfectly symmetrical shapes, meaning they look the same from any angle. This symmetry makes circles useful in many areas of mathematics and science. The chapter begins by exploring the basic definitions and properties of circles, including their symmetries and how to construct circles through given points.
A circle is defined as the set of all points on a plane that are equidistant from a given point (the centre).
The distance from the centre to any point on the circle is called the radius.
Circles have complete rotational symmetry, meaning they look the same after any rotation.
02
Chords, Arcs, and Angles in Circles
A chord is a line segment whose endpoints lie on the circle. The angle subtended by a chord at the centre is the angle formed by the radii drawn to the endpoints of the chord. The chapter explores the relationship between the length of a chord and the angle it subtends at the centre. It also introduces the concept of an arc, which is a connected portion of the circle defined by two points on the circle. The angle subtended by an arc at the centre is the angle formed by the radii drawn to the endpoints of the arc.
A chord is a line segment joining two points on the circle.
The angle subtended by a chord at the centre is equal to the angle formed by the radii drawn to the endpoints of the chord.
An arc is a connected portion of the circle defined by two points on the circle.
03
Cyclic Quadrilaterals and Their Properties
A quadrilateral is called cyclic if all its vertices lie on a single circle. The chapter explores the properties of cyclic quadrilaterals, including the fact that the sum of the measures of opposite angles of a cyclic quadrilateral is 180 degrees. This property is useful in solving problems involving cyclic quadrilaterals and in proving other geometric theorems.
A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle.
The sum of the measures of opposite angles of a cyclic quadrilateral is 180 degrees.
If a line segment between two points subtends equal angles at two other points on the same side of the segment, all four points lie on a single circle (i.e., the points are concyclic).
04
Tangents to Circles and Secants
A tangent to a circle is a line that touches the circle at exactly one point, called the point of tangency. The chapter explores the properties of tangents, including the fact that the tangent at any point of a circle is perpendicular to the radius drawn to the point of tangency. A secant is a line that intersects the circle at two points. The chapter also discusses the relationship between tangents and secants and their applications in geometry.
A tangent to a circle is a line that touches the circle at exactly one point.
The tangent at any point of a circle is perpendicular to the radius drawn to the point of tangency.
A secant is a line that intersects the circle at two points.
05
Circumference, Area, and Arc Length
The circumference of a circle is the distance around the circle, while the area is the space enclosed within the circle. The chapter provides formulas for calculating the circumference and area of a circle, as well as the length of an arc. These formulas are essential for solving problems involving circles and for understanding the applications of circles in various fields.
The circumference of a circle is given by the formula C = 2πr, where r is the radius.
The area of a circle is given by the formula A = πr².
The length of an arc is given by the formula L = rθ, where θ is the central angle in radians.
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What is the definition of a circle according to the chapter?
AThe set of all points on a plane that are equidistant from a given point on that plane.
BThe set of all points on a plane that are at a fixed distance from a fixed line.
CThe set of all points on a plane that are equidistant from two given points.
DThe set of all points on a plane that are at a constant distance from a fixed point outside the plane.
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Answer: (A) The set of all points on a plane that are equidistant from a given point on that plane.
The chapter defines a circle as the set of all points on the plane that are equidistant from a given point, which is the centre.
Question 02
What do we call a straight line that connects two points on a circle?
AChord
BRadius
CDiameter
DArc
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Answer: (A) Chord
A chord is a straight line segment that joins any two points on the circle. The radius connects the center to a point on the circle, and a diameter is a special chord passing through the center.
Question 03
A quadrilateral ABCD is cyclic. If ∠A = 80° and ∠B = 110°, what are the measures of ∠C and ∠D respectively?
A100° and 70°
B80° and 110°
C110° and 80°
D70° and 100°
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Answer: (A) 100° and 70°
In a cyclic quadrilateral, opposite angles sum to 180°. So, ∠C = 180° - ∠A = 100° and ∠D = 180° - ∠B = 70°.
Question 04
What is the area of a circle with radius 7 cm? (Use π = 22/7)
A44 cm²
B154 cm²
C616 cm²
D22 cm²
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Answer: (B) 154 cm²
Area = πr² = (22/7) × 7 × 7 = 154 cm².
Question 05
Which of the following statements about the symmetries of a circle is true?
AA circle has only one line of reflection symmetry.
BA circle has rotational symmetry only for angles that are multiples of 90°.
CA circle has complete rotational symmetry: rotating it by any angle makes it look exactly the same.
DA circle has no reflection symmetry.
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Answer: (C) A circle has complete rotational symmetry: rotating it by any angle makes it look exactly the same.
The chapter states that a circle has complete rotational symmetry because rotating it by any angle yields the same appearance.
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Q1. Define a circle. What are the centre, radius, chord, and diameter of a circle?
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Model answer
A circle is the set of all points on a plane that are equidistant from a given point on that plane. The given point is called the centre. The distance from the centre to any point on the circle is the radius. A line segment joining two points on the circle is called a chord. A chord passing through the centre is called a diameter.
Sample question3 marks
Q2. In a circle with centre O, chord AB subtends an angle of 60° at the centre. If the radius of the circle is 12 cm, find the length of the chord AB.
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Model answer
In triangle OAB, OA = OB = 12 cm (radii). Since ∠AOB = 60°, the triangle is equilateral (as two sides equal and included angle 60°). Therefore, AB = OA = OB = 12 cm. Hence, the length of chord AB is 12 cm.
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Q3. State the property of a cyclic quadrilateral regarding its opposite angles. If one angle of a cyclic quadrilateral is 75°, what is the measure of its opposite angle?
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Model answer
In a cyclic quadrilateral, the sum of the measures of each pair of opposite angles is 180°. Therefore, if one angle is 75°, its opposite angle is 180° − 75° = 105°.
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Q4. Define a tangent to a circle and a secant. How many tangents can be drawn to a circle from a point outside the circle?
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Model answer
A tangent to a circle is a line that touches the circle at exactly one point, called the point of contact. A secant is a line that intersects the circle at two distinct points. From a point outside the circle, exactly two tangents can be drawn to the circle.
Sample question3 marks
Q5. What is the circumference of a circle with radius 7 cm? (Use π = 22/7)
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Model answer
The circumference of a circle is given by the formula C = 2πr, where r is the radius. Substituting r = 7 cm and π = 22/7, we get C = 2 × (22/7) × 7 = 44 cm. Therefore, the circumference is 44 cm.
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