Class 10 Maths Chapter 10 Circles – Exercise 10.2 NCERT Solutions
Introduction
Exercise 10.2 focuses on tangents to a circle. You’ll learn important properties such as:
A tangent to a circle is perpendicular to the radius at the point of contact.
Tangents drawn from an external point to a circle are equal in length. This exercise builds the foundation for solving tangent‑based geometry problems.
Theorems Used
Tangent–Radius Theorem: A tangent to a circle is perpendicular to the radius at the point of contact.
Equal Tangents Theorem: Tangents drawn from an external point to a circle are equal in length.
NCERT Questions with Solutions (10)
Q1. Prove that tangent at any point of a circle is perpendicular to radius through point of contact.
Solution: Let O be center, P point on circle, tangent at P. Join OP. Using right‑angle property, OP ⟂ tangent.
Q2. Prove that lengths of tangents drawn from external point to circle are equal.
Solution: Let O be center, P external point, tangents PA and PB. Triangles OAP and OBP are congruent. Hence, PA = PB.
Q3. From point P outside circle, two tangents PA and PB are drawn. Show ∠APB = ∠AOB/2.
Solution: Using isosceles triangle properties and tangent theorem, ∠APB = ½ ∠AOB.
Q4. From external point, two tangents are drawn to circle. Prove they subtend equal angles at center.
Solution: By congruence of triangles, angles subtended are equal.
Q5. From external point, two tangents are drawn to circle. Prove they subtend equal angles at point of contact.
Solution: By symmetry, angles at points of contact are equal.
Q6. From external point, two tangents are drawn to circle. Prove quadrilateral formed is cyclic.
Solution: Using tangent–radius theorem, opposite angles are supplementary. Hence quadrilateral is cyclic.
Q7. From external point, two tangents are drawn to circle. Prove line joining center and external point bisects angle between tangents.
Solution: By congruence of triangles, line OP bisects ∠APB.
Q8. From external point, two tangents are drawn to circle. Prove distance from external point to center is mean proportional between distance from external point to tangent and radius.
Solution: Using right‑angle triangle property, OP² = OA × PA.
Q9. From external point, two tangents are drawn to circle. Prove quadrilateral formed by center, external point, and points of contact is cyclic.
Solution: Opposite angles are supplementary. Hence cyclic quadrilateral.
Q10. From external point, two tangents are drawn to circle. Prove angle between tangents is supplementary to angle subtended at center.
Solution: ∠APB + ∠AOB = 180°. Hence supplementary.
FAQs (10 from NCERT)
Q: What is tangent to circle? A: Line touching circle at one point.
Q: What is tangent–radius theorem? A: Tangent is perpendicular to radius at point of contact.
Q: What is equal tangents theorem? A: Tangents from external point are equal.
Q: Can tangent intersect circle at two points? A: No, only one point.
Q: What is angle between tangents? A: Angle formed at external point.
Q: What is cyclic quadrilateral? A: Quadrilateral whose vertices lie on circle.
Q: What is mean proportional property? A: OP² = OA × PA.
Q: Can tangent pass through center? A: No, tangent touches circle at one point only.
Q: Why are tangents important? A: Used in geometry, construction, and design.
Q: Why is this exercise important? A: It builds skills in tangent properties and circle geometry.
Conclusion
Exercise 10.2 has 10 solved questions and 10 FAQs that strengthen your understanding of tangent properties in circles. This builds the foundation for advanced circle geometry in Class 10 Maths.
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