Circles Class 9 NCERT Solutions for comprehensive understanding. Get step-by-step explanations and practice problems to improve your geometry skills for exams.
Circles Class 9 NCERT Solutions Questions and Answers
1. What is the definition of a circle in geometry?
A circle is a set of points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius.
2. How can you define the radius and diameter of a circle?
The radius of a circle is the distance from the center of the circle to any point on the circle. The diameter is twice the length of the radius and passes through the center of the circle.
3. What is the circumference of a circle?
The circumference of a circle is the perimeter or boundary of the circle. It is calculated as 2π times the radius or π times the diameter.
4. What does the term 'chord' mean in a circle?
A chord is a line segment joining any two points on the circle. The longest chord in any circle is the diameter.
5. Can you explain what a tangent to a circle is?
A tangent is a straight line that touches the circle at exactly one point. This point is known as the point of contact.
6. What is the meaning of the term 'secant' in relation to a circle?
A secant is a line that intersects the circle at two distinct points.
7. How is an arc defined in a circle?
An arc is a portion of the circumference of a circle, determined by two points on the circle. The length of the arc depends on the angle subtended by it at the center of the circle.
8. What is the sector of a circle?
A sector is the region enclosed by two radii and the arc between them. It resembles a 'pie slice' of the circle.
9. Can you explain the central angle of a circle?
The central angle of a circle is the angle formed at the center of the circle by two radii. The measure of the central angle is directly related to the size of the arc it subtends.
10. What is the angle subtended by a diameter at any point on the circle?
The angle subtended by a diameter at any point on the circle is always a right angle, i.e., 90 degrees.
11. How do you calculate the length of an arc of a circle?
The length of an arc can be calculated using the formula: (θ/360) × 2πr, where θ is the central angle in degrees and r is the radius.
12. What is the area of a sector of a circle?
The area of a sector is calculated using the formula: (θ/360) × πr², where θ is the central angle and r is the radius.
13. What is the area of a circle?
The area of a circle is calculated as πr², where r is the radius of the circle.
14. How can you find the length of a chord in a circle?
To find the length of a chord, you can use the relationship between the radius, the distance from the center to the chord, and the length of the chord.
15. What is the perpendicular from the center to a chord?
The perpendicular from the center of a circle to a chord bisects the chord, dividing it into two equal parts.
16. What is the relationship between the radius and the tangent at a point on the circle?
The radius at the point of contact of a tangent is perpendicular to the tangent. This means the angle between the radius and the tangent is always 90 degrees.
17. How is the circle divided by two perpendicular chords?
Two perpendicular chords of a circle divide the circle into four regions or segments. Each of these regions can be a sector or an arc.
18. Can the angle formed between two tangents to a circle be greater than 180 degrees?
No, the angle formed between two tangents drawn from an external point to a circle is always less than or equal to 180 degrees.
19. What is the relation between the angle subtended by an arc at the center and at the circumference?
The angle subtended by an arc at the center is twice the angle subtended at the circumference of the circle.
20. Can you explain the property of cyclic quadrilaterals in a circle?
A cyclic quadrilateral is a quadrilateral whose vertices lie on a circle. The sum of the opposite angles of a cyclic quadrilateral is always 180 degrees.
21. What is the power of a point in relation to a circle?
The power of a point with respect to a circle is the square of the distance from the point to the center of the circle minus the square of the radius.
22. How can the equation of a circle be represented in a coordinate plane?
The standard equation of a circle in a coordinate plane is given by (x-h)² + (y-k)² = r², where (h,k) is the center and r is the radius.
23. How can you prove that the tangent to a circle is perpendicular to the radius?
One can prove this by using the property of the angle formed between a radius and a tangent being 90 degrees at the point of contact.
24. What is the significance of the angle of 180 degrees in a circle?
An angle of 180 degrees in a circle corresponds to a straight line, which is the diameter of the circle.
25. Can you define a segment in a circle?
A segment in a circle is the region bounded by a chord and the arc subtended by it.
26. What happens when two tangents are drawn from an external point to a circle?
The two tangents drawn from an external point to a circle are equal in length, and the angle between them is always the same.
27. How can you find the distance between two points on a circle?
The distance between two points on a circle can be found using the length of the chord joining those points and the radius of the circle.
28. What is the effect of a chord being closer to the center of a circle?
The closer a chord is to the center of the circle, the longer it becomes. Conversely, the farther it is from the center, the shorter the chord.
29. Can a tangent ever intersect the circle?
No, a tangent only touches the circle at one point and does not intersect it at any other point.
30. How do you find the area of a circular ring?
The area of a circular ring is found by subtracting the area of the smaller circle from the area of the larger circle.
31. What is the formula for finding the circumference of a circle?
The formula to calculate the circumference of a circle is 2πr, where r is the radius of the circle.
32. Can a tangent to a circle be drawn at any point?
Yes, a tangent can be drawn at any point on the circle. There is exactly one tangent at every point on the circle.
33. What is the relationship between the length of a chord and the distance from the center to the chord?
The shorter the distance from the center to the chord, the longer the chord. If the distance is zero (i.e., the perpendicular from the center to the chord), the chord is the diameter of the circle.
34. What is the relationship between the radius and the chord when the chord passes through the center?
When the chord passes through the center of the circle, it is the diameter, and the radius is half of the length of the diameter.
35. How can you describe the point where two tangents to a circle meet?
The point where two tangents meet is called the point of tangency. This point lies on the circle, and the tangents make equal angles with the line joining the point of tangency to the center.
36. Can you explain how to find the center of a circle using two chords?
To find the center of a circle, one can draw perpendicular bisectors to two chords. The point where these bisectors meet is the center of the circle.
37. How do you calculate the angle between two intersecting chords?
The angle between two intersecting chords can be calculated using the intersection point and the properties of the angles formed between the chords.
38. What is the relationship between the center of the circle and the perpendicular bisector of a chord?
The perpendicular bisector of any chord passes through the center of the circle.
39. How is the central angle related to the arc it subtends?
The central angle determines the length of the arc it subtends. A larger central angle subtends a longer arc.
40. What is the meaning of the term 'sector angle'?
The sector angle is the central angle that defines the extent of the sector in a circle.
41. What is the relationship between the length of a chord and the size of the arc it subtends?
The longer the chord, the larger the arc it subtends, which corresponds to a larger central angle.
42. How does the area of a sector change with the size of the central angle?
The area of a sector increases as the size of the central angle increases. The area is directly proportional to the central angle.
43. What is the effect of an inscribed angle in a circle?
An inscribed angle in a circle is half the measure of the central angle subtended by the same arc.
44. Can you explain the concept of a cyclic polygon?
A cyclic polygon is any polygon whose vertices lie on a circle. The sum of the opposite angles of a cyclic quadrilateral is always 180 degrees.
45. What is the significance of the point of intersection of two tangents to a circle?
The point where two tangents meet is important in geometry because it forms an angle equal to half the difference between the angles subtended by the tangents.
46. What is the relationship between the radius of a circle and the angle subtended at the center by a chord?
The greater the radius of the circle, the greater the angle subtended at the center by any chord, assuming the length of the chord remains constant.
47. How can the area of a circular sector be used to solve real-world problems?
The area of a circular sector can be applied to real-world scenarios like calculating the area of slices of pizza, fan blades, or segments of a clock.
48. How can you find the length of a chord when you know the radius and the perpendicular distance from the center to the chord?
You can apply the Pythagorean theorem to find the length of a chord by using the radius and the perpendicular distance from the center to the chord.
49. What is the relationship between the arc of a circle and its chord?
The length of the chord is related to the length of the arc, with the chord being shorter than the arc when the central angle is less than 180 degrees.
50. What happens when a chord is equal in length to the radius of a circle?
If a chord is equal in length to the radius of a circle, the perpendicular from the center to the chord bisects the chord and divides the circle into two equal parts.
Top Indian Books for Circles Class 9 NCERT Solutions
"Mathematics for Class 9" by R.D. Sharma (Dhanpat Rai Publications)
This book offers detailed explanations of the concepts in the chapter "Circles." It covers all important definitions, theorems, and their proofs. The exercises are designed to test comprehension, with questions ranging from basic to complex.
"Comprehensive Mathematics for Class 9" by R.S. Aggarwal (S. Chand Publications)
The book provides step-by-step solutions for all the exercises. It focuses on helping students grasp the underlying principles of geometry, with clear explanations of tangents, chords, and sectors, as well as a variety of practice questions.
"Understanding Mathematics Class 9" by M.L. Agarwal (Arihant Publications)
Focusing on the topic of circles, this book offers both theoretical content and practical examples. The questions range from simple to advanced, helping students develop problem-solving skills while covering all necessary concepts.
"Mathematics for Class 9" by R.S. Aggarwal (S. Chand Publications)
This book covers the entire curriculum of Class 9 Mathematics. For the "Circles" chapter, it includes multiple types of problems: from identifying geometrical properties to solving real-world applications involving circles.
"New Pattern Mathematics for Class 9" by J.L. Arora (S. Chand Publications)
This book emphasizes pattern recognition in geometric concepts. It offers multiple examples and problem sets for the circles chapter, focusing on the application of formulas in various geometrical problems.
"Mathematics for Class 9" by S.K. Gupta and Anubhuti Gangal (Laxmi Publications)
The book is well-organized with clear explanations of theorems and concepts related to circles. The variety of problems provides extensive practice for students, especially on topics like tangents, sectors, and cyclic quadrilaterals.
"Mathematics Class 9" by V.K. Agarwal (VK Publications)
With a focus on the NCERT syllabus, this book offers clear, step-by-step solutions for all exercises in the "Circles" chapter. It includes various types of questions and is especially helpful for exam preparation.
"Mathematics Class 9" by R.K. Bansal (Universal Book Stall)
This book is designed for a thorough understanding of the geometry of circles. It provides a variety of practice problems, including theoretical and application-based questions.
"Objective Mathematics for Class 9" by R.D. Sharma (Dhanpat Rai Publications)
With a strong focus on objective questions, this book is ideal for students preparing for competitive exams. It includes short-answer questions, true or false questions, and more, which test knowledge of key concepts related to circles.
"Quick Learning Mathematics Class 9" by S.K. Sharma (Dhanpat Rai Publications)
This is a student-friendly book with an easy-to-understand approach. The chapters are well-organized with a focus on key concepts and formulae. It includes both solved and unsolved exercises that promote deeper learning.
"Mathematics for Class 9" by P. K. Mittal (Ratan Publications)
The book breaks down the "Circles" chapter with clear definitions, properties, and plenty of solved examples. It is designed for students who need a comprehensive guide to understanding geometrical concepts related to circles.
"NCERT Exemplar Problems Class 9 Mathematics" (NCERT Publications)
This book offers challenging and application-based problems, helping students hone their understanding of circles. It provides the perfect mix of theory and problem-solving practice.
"Mathematics Class 9" by R.K. Jain (Khanna Publishers)
This book dives deep into geometric concepts, particularly circles. It provides detailed proofs, explanations, and ample practice questions to ensure a thorough understanding of the subject.
"Mathematics Class 9" by T.K. Rengarajan (Pragati Prakashan)
Ideal for exam-focused students, this book includes a variety of problems from the "Circles" chapter, from easy to moderate difficulty levels. It ensures a complete grasp of both basic and advanced concepts in circle geometry.
"Concise Mathematics Class 9" by R.S. Aggarwal (S. Chand Publications)
Concise and to the point, this book provides a detailed explanation of each topic in the "Circles" chapter. It includes numerous practice questions, with solutions provided for better clarity.
"M.K. Singhal’s Mathematics for Class 9" by M.K. Singhal (Singhal Publications)
This book presents the concepts of circles with clear definitions and theorems, making it easier for students to understand. It has practical problems to work through, with answers at the end of each chapter.
"Mathematics for Class 9" by R.S. Aggarwal and V. Aggarwal (S. Chand Publications)
The book is excellent for detailed learning and problem practice. It includes multiple-choice questions, short-answer questions, and long-answer questions on the topic of circles.
"Target Class 9 Mathematics" by S.K. Kapoor (Arihant Publications)
This book offers detailed solutions to each chapter, with a special focus on geometry problems related to circles. The book includes objective-type questions that help students prepare for competitive exams.
"Modern's ABC of Mathematics Class 9" by Modern Publishers
This book emphasizes geometry with an in-depth study of circles. With multiple examples and practice sets, it allows students to enhance their understanding of key concepts and improve their problem-solving skills.
"Foundation Mathematics for Class 9" by G. K. Publications
Focused on building a strong foundation, this book covers the essential concepts related to circles. It includes simple, clear explanations and plenty of practice exercises to reinforce the learning.
Circles Class 9 NCERT Solutions: Understanding the Basics of Geometry
Circles are one of the most fascinating and essential concepts in geometry. For Class 9 students, the study of circles forms a significant portion of their Mathematics syllabus. The "Circles" chapter in NCERT solutions covers a range of topics such as the properties of circles, tangents, chords, secants, and various theorems related to these concepts.
The NCERT solutions for Class 9 include a wide range of problems that test a student's ability to apply geometric principles in real-life scenarios. These problems typically involve understanding the various properties of circles, such as the relation between a tangent and a radius, the calculation of the length of an arc or sector, and finding angles subtended by different elements of a circle. The exercises include both simple theoretical questions and more complex problem-solving ones that encourage students to think critically.
One of the key areas covered in the "Circles" chapter is the theorem stating that the angle subtended by a diameter of a circle at any point on the circle is always a right angle. This theorem is fundamental in solving various types of problems and is frequently tested in exams. The chapter also delves into other important theorems, including the angle subtended by an arc at the center being twice the angle subtended at the circumference, which plays a crucial role in solving geometry-based questions.
Another critical part of the chapter involves understanding the properties of tangents. For example, the property that the tangent to a circle at any point is perpendicular to the radius drawn to the point of contact is essential for answering questions related to tangents and their lengths. Problems involving tangents often require students to apply their knowledge of trigonometry and algebra.
Additionally, students are introduced to the concept of cyclic quadrilaterals, which are quadrilaterals whose vertices lie on the circle. The chapter offers a variety of exercises on cyclic quadrilaterals, helping students understand their properties and the relationship between their angles.
In conclusion, the "Circles" chapter in Class 9 is an integral part of the Mathematics curriculum. It lays the foundation for more advanced topics in geometry that students will encounter in higher classes. Mastering the concepts in this chapter is essential for excelling in both school exams and competitive tests. The NCERT solutions provide students with the necessary tools to understand, apply, and solve problems related to circles, making it an indispensable resource for any student looking to master this fundamental topic in geometry.
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