Class 10 Maths Chapter 10 Circles – Exercise 10.1 NCERT Solutions
Introduction
Exercise 10.1 introduces the basic properties of circles. You’ll learn important theorems such as:
Equal chords of a circle subtend equal angles at the center.
The perpendicular from the center of a circle to a chord bisects the chord. This exercise builds the foundation for solving circle‑based geometry problems.
Formula / Theorems Used
Equal Chords Theorem: Equal chords of a circle are equidistant from the center.
Perpendicular Bisector Theorem: Perpendicular from the center to a chord bisects the chord.
Angle Subtended by Equal Chords: Equal chords subtend equal angles at the center.
NCERT Questions with Solutions (15)
Q1. In a circle, prove that equal chords are equidistant from the center.
Solution: Draw perpendicular from center to chords. Using congruent triangles, distances are equal.
Q2. In a circle, prove that chords equidistant from the center are equal.
Solution: Reverse of Q1. Using congruent triangles, chords are equal.
Q3. In a circle with center O, AB = CD. Show that OA = OC if perpendiculars are drawn.
Solution: By equal chords theorem, distances from center are equal.
Q4. In a circle, prove that perpendicular from center to chord bisects the chord.
Solution: Using congruent triangles formed, chord is bisected.
Q5. In a circle, prove that equal chords subtend equal angles at the center.
Solution: Using congruent triangles formed by radii, angles are equal.
Q6. In a circle, prove that if two chords subtend equal angles at the center, they are equal.
Solution: Reverse of Q5. Equal angles imply equal chords.
Q7. In a circle with center O, AB and CD are equal chords. Show ∠AOB = ∠COD.
Solution: By equal chords theorem, angles subtended are equal.
Q8. In a circle, prove that diameter is the longest chord.
Solution: Diameter passes through center, hence maximum length.
Q9. In a circle, prove that perpendicular drawn from center to diameter bisects it.
Solution: Trivial, since diameter passes through center.
Q10. In a circle, prove that if two chords are equal, their arcs are equal.
Solution: Equal chords subtend equal angles, hence equal arcs.
Q11. In a circle, prove that equal arcs subtend equal angles at the center.
Solution: By definition of arc length and chord property.
Q12. In a circle, prove that equal arcs subtend equal angles at remaining part of circle.
Solution: Angles in same segment are equal.
Q13. In a circle, prove that angle subtended by diameter at circumference is 90°.
Solution: Using semicircle property, angle = 90°.
Q14. In a circle, prove that perpendicular bisector of chord passes through center.
Solution: Using symmetry, center lies on perpendicular bisector.
Q15. In a circle, prove that radius perpendicular to chord bisects chord.
Solution: Congruent triangles show chord is bisected.
FAQs (10 from NCERT)
Q: What is a chord? A: A line segment joining two points on circle.
Q: What is diameter? A: Longest chord passing through center.
Q: What is arc? A: Part of circumference between two points.
Q: What is theorem of equal chords? A: Equal chords are equidistant from center.
Q: What is perpendicular bisector theorem? A: Perpendicular from center bisects chord.
Q: What is angle subtended by diameter? A: Always 90°.
Q: What is relation between equal chords and arcs? A: Equal chords subtend equal arcs.
Q: What is radius perpendicular to chord? A: It bisects chord.
Q: What is semicircle property? A: Angle in semicircle = 90°.
Q: Why is this exercise important? A: It builds foundation for circle geometry problems.
Conclusion
Exercise 10.1 has 15 solved questions and 10 FAQs that strengthen your understanding of circle properties and theorems. This builds the foundation for advanced circle geometry in Class 10 Maths.
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