Class 9 Maths Chapter 9 Circles – Exercise 9.4 NCERT Solutions

Introduction

Exercise 9.4 introduces the tangent properties of circles. You’ll learn how to prove that the tangent at any point of a circle is perpendicular to the radius at that point, and apply this property to solve problems involving tangents.

Key Concept

• Tangent–Radius Property: The tangent at any point of a circle is perpendicular to the radius through that point.

$$OP\perp PT$$

where O is the center, P is the point of contact, and PT is the tangent.

Solved Questions (Step by Step)

Q1. Prove that tangent at any point of circle is perpendicular to radius.

Solution:

• Let O be center, P point of contact, PT tangent.

• OP ⟂ PT.

Q2. In circle with center O, tangent at A touches circle. Prove OA ⟂ tangent.

Solution:

• Radius at point of contact is perpendicular to tangent.

Q3. In circle with center O, tangent at B touches circle. Prove OB ⟂ tangent.

Solution:

• Radius at point of contact is perpendicular to tangent.

Q4. In circle with center O, tangent at C touches circle. Prove OC ⟂ tangent.

Solution:

• Radius at point of contact is perpendicular to tangent.

Q5. In circle with center O, tangent at D touches circle. Prove OD ⟂ tangent.

Solution:

• Radius at point of contact is perpendicular to tangent.

Q6. In circle with center O, tangent at E touches circle. Prove OE ⟂ tangent.

Solution:

• Radius at point of contact is perpendicular to tangent.

Q7. In circle with center O, tangent at F touches circle. Prove OF ⟂ tangent.

Solution:

• Radius at point of contact is perpendicular to tangent.

Q8. In circle with center O, tangent at G touches circle. Prove OG ⟂ tangent.

Solution:

• Radius at point of contact is perpendicular to tangent.

Q9. In circle with center O, tangent at H touches circle. Prove OH ⟂ tangent.

Solution:

• Radius at point of contact is perpendicular to tangent.

Q10. In circle with center O, tangent at I touches circle. Prove OI ⟂ tangent.

Solution:

• Radius at point of contact is perpendicular to tangent.

Q11. In circle with center O, tangent at J touches circle. Prove OJ ⟂ tangent.

Solution:

• Radius at point of contact is perpendicular to tangent.

Q12. In circle with center O, tangent at K touches circle. Prove OK ⟂ tangent.

Solution:

• Radius at point of contact is perpendicular to tangent.

Q13. In circle with center O, tangent at L touches circle. Prove OL ⟂ tangent.

Solution:

• Radius at point of contact is perpendicular to tangent.

Q14. In circle with center O, tangent at M touches circle. Prove OM ⟂ tangent.

Solution:

• Radius at point of contact is perpendicular to tangent.

Q15. In circle with center O, tangent at N touches circle. Prove ON ⟂ tangent.

Solution:

• Radius at point of contact is perpendicular to tangent.

FAQs (10 for Exercise 10.4)

1. Q: What is a tangent to a circle? A: A line that touches the circle at exactly one point.

2. Q: What is the tangent–radius property? A: Tangent at point of contact is perpendicular to radius.

3. Q: How many tangents can be drawn to a circle? A: Infinitely many.

4. Q: How many tangents can be drawn from an external point? A: Two tangents.

5. Q: What is the difference between chord and tangent? A: Chord intersects circle at two points; tangent touches at one point.

6. Q: Why is tangent perpendicular to radius? A: Because radius is shortest distance from center to tangent.

7. Q: Can tangent pass through center? A: No, that would be a diameter.

8. Q: What is the angle between radius and tangent? A: 90°.

9. Q: Why are tangent properties important? A: They help in solving geometry problems and constructions.

10. Q: Can tangent intersect circle at more than one point? A: No, only one point.

Conclusion

Exercise 9.4 has 15 questions that strengthen your understanding of tangent properties of circles. This completes Chapter 10 of Class 9 Maths.

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