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Class 10 Mathematics -> Chapter 10: Circle
I. Chapter Summary:
This chapter focuses on the properties and theorems related to circles. A circle is a set of points that are equidistant from a fixed point called the center. Students will learn about key elements of a circle, including the radius, diameter, and chord. The chapter explores important theorems such as the angle subtended by a chord at the center of the circle, the relationship between tangents and radii, and the cyclic quadrilaterals. Additionally, the chapter covers the construction of tangents and problems involving tangents, secants, and chords.
II. Key Concepts Covered:
- Basic Elements of a Circle:
- A circle is the locus of all points at a fixed distance (radius) from a fixed point (center).
- Radius: The distance from the center to any point on the circle.
- Diameter: The longest chord of the circle, passing through the center.
- Chord: A line segment joining any two points on the circle.
- Secant: A line that intersects the circle at two points.
- Tangent: A line that touches the circle at exactly one point.
- Theorems Related to Circles:
- Theorem 1: The angle subtended by a chord at the center of the circle is twice the angle subtended at any point on the remaining part of the circle.
- Theorem 2: Angles subtended by the same chord at the circumference are equal.
- Theorem 3: The perpendicular from the center of the circle to a chord bisects the chord.
- Theorem 4: The lengths of two tangents drawn from an external point to a circle are equal.
- Tangent and Its Properties:
- A tangent to a circle is perpendicular to the radius drawn to the point of contact.
- Two tangents from an external point to a circle are equal in length.
- Cyclic Quadrilaterals:
- A cyclic quadrilateral is a quadrilateral where all four vertices lie on the circle.
- Opposite angles of a cyclic quadrilateral are supplementary.
- Constructing Tangents:
- The process of constructing a tangent from an external point to a circle is covered in the chapter.
III. Important Questions:
(A) Multiple Choice Questions (MCQs) (1 Mark):
- The angle subtended by a chord at the center of the circle is:
- (A) Half the angle subtended at the circumference
- (B) Equal to the angle subtended at the circumference
- (C) Twice the angle subtended at the circumference
- (D) None of the above
- Answer: (C)
- The tangents from an external point to a circle are:
- (A) Equal in length
- (B) Unequal in length
- (C) Parallel
- (D) Perpendicular
- Answer: (A)
- In a circle, the sum of the opposite angles of a cyclic quadrilateral is:
- (A) 90°
- (B) 180°
- (C) 360°
- (D) 270°
- Answer: (B)
- The perpendicular from the center of the circle to a chord:
- (A) Divides the chord in any ratio
- (B) Does not bisect the chord
- (C) Bisects the chord
- (D) Is not perpendicular to the chord
- Answer: (C)
(B) Short Answer Questions (2/3 Marks):
- If the angle subtended by a chord at the center of a circle is 80°, find the angle subtended by the same chord at any point on the remaining part of the circle.
- In a circle, the length of the chord is 10 cm, and the perpendicular distance from the center of the circle to the chord is 6 cm. Find the radius of the circle.
- Prove that the tangents drawn to a circle from an external point are equal in length.
- In a cyclic quadrilateral, if one angle is 120°, find the other angle formed by the opposite side.
(C) Long Answer Questions (5 Marks):
- Prove that the angle between the tangent and the radius at the point of contact is 90°.
- In a circle with center O, two chords AB and CD intersect at P. Prove that $angle APD + angle BPC = 180^circ$
- A chord AB is at a distance of 5 cm from the center O of the circle with a radius of 13 cm. Find the length of the chord AB.
- In a cyclic quadrilateral ABCD, if $angle A + angle C = 180^circ$, prove that the quadrilateral is cyclic.
(D) HOTS (Higher Order Thinking Skills) Questions:
- A circle with a radius of 10 cm has two tangents drawn from an external point. The distance between the external point and the center of the circle is 15 cm. Find the length of the tangents.
- In a circle, a chord of length 12 cm is 5 cm away from the center. Find the radius of the circle and the angles subtended by the chord at the center.
IV. Key Formulas/Concepts:
- Length of Tangents:
- If a tangent is drawn from an external point P to a circle with center O and radius r, and the distance from P to the center O is d, then the length of the tangent t is given by:
$t = sqrt{d^2 – r^2}$
- Theorem on Angles Subtended by a Chord:
- The angle subtended by a chord at the center is twice the angle subtended at any point on the remaining part of the circle.
- If (theta) is the angle at the center and (alpha) is the angle at the circumference, then:
$theta = 2 times alpha$
- Cyclic Quadrilateral:
-
Opposite angles of a cyclic quadrilateral are supplementary, i.e.:
$angle A + angle C = 180^circ, quad angle B + angle D = 180^circ$
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V. Deleted Portions (CBSE 2025-2026):
No portions have been deleted from this chapter as per the rationalized NCERT textbooks.
VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025-2026):
| Unit/Chapter | Estimated Marks | Type of Questions Typically Asked |
|---|---|---|
| Circle | 6 | 1 Mark (MCQs), 2/3 Marks (Short Answers), 5 Marks (Long Answers) |
VII. Previous Year Questions (PYQs):
- 2019:
- (1 Mark) If two tangents are drawn from a point outside a circle, what is the angle between the two tangents?
- (2 Marks) Prove that the lengths of two tangents drawn from an external point to a circle are equal.
- (5 Marks) In a circle, the angle subtended by a chord at the center is 60°. Find the angle subtended by the chord at any point on the remaining part of the circle.
- 2020:
- (3 Marks) Prove that the perpendicular from the center of a circle to a chord bisects the chord.
- (5 Marks) A circle has a radius of 13 cm. Find the distance from the center of the circle to a chord of length 10 cm.
VIII. Real-World Application Examples to Connect with Topics:
- Engineering and Architecture:
- The concepts of tangents and chords are used in the design of circular structures such as domes, bridges, and arches.
- Navigation and Mapping:
- Trigonometric ratios and circle properties are used in navigation to find distances between locations on maps, especially when dealing with circular or spherical regions.
- Astronomy:
- In astronomy, circles and their properties are used to calculate the positions of celestial bodies and the distances between them.
IX. Student Tips & Strategies for Success:
- Time Management:
- Focus on understanding theorems and their proofs. Regular practice will help improve speed and accuracy, especially in word problems involving circles.
- Exam Preparation:
- Practice solving problems on tangents, cyclic quadrilaterals, and angles subtended by chords. Focus on mastering geometric proofs and constructions.
- Stress Management:
- Break down complex problems into smaller steps, and work on understanding the core principles. Take short breaks to avoid burnout.
X. Career Guidance & Exploration (Class-Specific):
- For Classes 9-10:
- Understanding circle properties is essential for students aspiring to pursue careers in Engineering, Architecture, and Astronomy. These topics also form the foundation for competitive exams like JEE.
- For Classes 11-12:
- The knowledge of circles is crucial for entrance exams like JEE, NEET, and NDA, especially for careers in Mechanical Engineering, Physics, Mathematics, and Architecture.
XI. Important Notes:
- Regular revision of theorems, formulas, and practice of construction problems will help solidify your understanding.
- Refer to the official CBSE website for updates on the syllabus and exam guidelines.