Class 9 Maths Chapter 9 Circles – Exercise 9.1 NCERT Solutions
Introduction
Exercise 10.1 introduces the fundamental properties of circles. You’ll learn how to prove that equal chords subtend equal angles at the center, the perpendicular from the center bisects the chord, and related theorems.
Key Concepts
Equal Chords Theorem: Equal chords of a circle subtend equal angles at the center.
Perpendicular Bisector Theorem: Perpendicular from the center of a circle to a chord bisects the chord.
Converse Theorem: If two chords subtend equal angles at the center, they are equal.
Solved Questions (Step by Step)
Q1. Prove that equal chords of a circle subtend equal angles at the center.
Solution:
Let AB and CD be equal chords.
Join OA, OB, OC, OD (O = center).
Triangles OAB and OCD are congruent.
Hence, ∠AOB = ∠COD.
Q2. Prove that if two chords subtend equal angles at the center, they are equal.
Solution:
Let ∠AOB = ∠COD.
Triangles OAB and OCD are congruent.
Hence, AB = CD.
Q3. Prove that perpendicular from center to chord bisects the chord.
Solution:
Let OP ⟂ AB.
Triangles OAP and OBP are congruent.
Hence, AP = PB.
Q4. Prove that equal chords are equidistant from the center.
Solution:
Let AB = CD.
Perpendicular distances OP and OQ from center are equal.
Q5. Prove that chords equidistant from center are equal.
Solution:
Let OP = OQ.
Triangles formed show AB = CD.
Q6. In circle with center O, AB = CD. Prove ∠AOB = ∠COD.
Solution:
By equal chords theorem, equal chords subtend equal angles.
Q7. In circle with center O, ∠AOB = ∠COD. Prove AB = CD.
Solution:
By converse theorem, equal angles subtend equal chords.
Q8. In circle with center O, OP ⟂ AB. Prove AP = PB.
Solution:
By perpendicular bisector theorem, chord is bisected.
Q9. In circle with center O, AB = CD. Prove OP = OQ.
Solution:
Equal chords are equidistant from center.
Q10. In circle with center O, OP = OQ. Prove AB = CD.
Solution:
Chords equidistant from center are equal.
Q11. In circle with center O, AB = CD. Prove ΔOAB ≅ ΔOCD.
Solution:
Equal chords subtend equal angles.
Triangles are congruent.
Q12. In circle with center O, ∠AOB = ∠COD. Prove ΔOAB ≅ ΔOCD.
Solution:
Equal angles subtend equal chords.
Triangles are congruent.
Q13. In circle with center O, OP ⟂ AB. Prove ΔOAP ≅ ΔOBP.
Solution:
By RHS congruence, ΔOAP ≅ ΔOBP.
Q14. In circle with center O, AB = CD. Prove ΔOAP ≅ ΔOCQ.
Solution:
Equal chords equidistant from center.
Triangles congruent.
Q15. In circle with center O, OP = OQ. Prove ΔOAP ≅ ΔOCQ.
Solution:
Chords equidistant from center are equal.
Triangles congruent.
FAQs (10 for Exercise 10.1)
Q: What is a chord of a circle? A: A line segment joining two points on the circle.
Q: What is the perpendicular bisector theorem? A: Perpendicular from center to chord bisects the chord.
Q: What is the equal chords theorem? A: Equal chords subtend equal angles at the center.
Q: What is the converse of equal chords theorem? A: If chords subtend equal angles, they are equal.
Q: What is the distance of a chord from the center? A: Length of perpendicular from center to chord.
Q: When are two chords equal? A: When they subtend equal angles at the center or are equidistant from center.
Q: What is the radius of a circle? A: Distance from center to any point on circle.
Q: What is the diameter of a circle? A: A chord passing through the center, twice the radius.
Q: Why are circle theorems important? A: They form the basis of geometry proofs and constructions.
Q: How many equal chords can a circle have? A: Infinitely many, depending on radius and position.
Conclusion
Exercise 9.1 has 15 questions that strengthen your understanding of circle properties, especially chords, angles, and perpendiculars. This builds the foundation for advanced circle theorems in later exercises.
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