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Download Free PDF of Circles Exercise 10.1 Solutions for Class 10 Maths
If you’re tackling Class 10 Maths, mastering the concepts in Chapter 10 Circles is crucial for your CBSE board exams. This chapter regularly carries a weightage of about 7 marks and builds your foundation in essential geometry, from the definition of tangents to proofs involving chords and circle properties.
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Here, you will find NCERT Solutions for Class 10 Maths Chapter 10 Circles Exercise 10.1, organized in stepwise methods. Each solution not only resolves core circle geometry problems, but also follows the format expected in board exams. You'll see how the radius is always perpendicular to a tangent and learn to recognize board exam questions, just like those seen for queries such as "class 10 circles exercise 10.1 solutions."
With support for circle theorem explanations and tips to avoid common mistakes, these practice solutions help you develop confidence for exam-style questions. All answers are checked and explained by Vedantu's experienced maths faculty, so you’re always preparing with accuracy and clarity.
CBSE Class 10 Mathematics NCERT Solutions Chapter 10 Circles – Free PDF Download 2025-26
ShareShareExercise 10.1
1.How many tangents can a circle have?
Ans: A circle can have an infinite number of tangents. A circle is made of infinite points and from each point, a tangent can be formed.
2. Fill in the blanks :
(i) A tangent to a circle intersects it in _________ point(s).
Ans: A tangent to a circle intersects it in exactly one point.
(ii) A line intersecting a circle in two points is called a _______.
Ans: A line intersecting a circle in two points is called a secant.
(iii) A circle can have _______ parallel tangents at the most.
Ans: A circle can have two parallel tangents at the most opposite to one another.
(iv) The common point of a tangent to a circle and the circle is called ______.
Ans: The common point of a tangent to a circle and the circle is called the Point of Contact.
3. A tangent PQ at a point P of a circle of radius \[\text{5cm}\] meets a line through the centre O at a point Q so that \[\text{OQ = 12cm}\]. Length of PQ is :
(A) \[\text{12}\]cm
(B) \[\text{13}\]cm
(C) \[\text{8}\text{.5}\] cm
(D)\[\sqrt{\text{119}}\]
Ans: According to the Pythagoras Theorem: \[PQ=\sqrt{\left( O{{Q}^{2}}-\left. O{{P}^{2}} \right) \right.}\]
OQ = \[\text{12}\]cm, OP = \[\text{5}\]cm
Because,
\[PQ=\sqrt{\left( O{{Q}^{2}}-\left. O{{P}^{2}} \right) \right.}\]
\[=\sqrt{\left( {{12}^{2}}-\left. {{5}^{2}} \right) \right.}\]
\[=\sqrt{144-25}\]
\[=\sqrt{119}\]cm.
Hence, (D) is the answer.
4. Draw a circle and two lines parallel to a given line such that one is tangent and the other, a secant to the circle.
Ans: From the Given Figure below,
Let ‘l’ be the given line and a circle with centre O is drawn.
A tangent to the circle from point B is drawn \[\parallel \] to line ‘l’.
CD is drawn \[\parallel \] to line ‘l’ and is the secant.
Conclusion
Vedantu's NCERT Maths Chapter 10 Class 10 Circles Exercise 10.1 Solutions includes detailed solutions and explanations to all questions. We will understand the properties of circles and tangents, especially the relationship between the radius and the tangent at the point of contact. By practising these exercises, we strengthen these concepts and learn the theorems about tangents. Ex 10.1 Class 10 solutions will help you develop an excellent foundation in circle geometry and prepare you for exams.
Class 10 Maths Chapter 10: Exercises Breakdown
CBSE Class 10 Maths Chapter 10 Other Study Materials
Chapter-Specific NCERT Solutions for Class 10 Maths
Given below are the chapter-wise NCERT Solutions for Class 10 Maths. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.
Study Resources for Class 10 Maths
For complete preparation of Maths for CBSE Class 10 board exams, check out the following links for different study materials available at Vedantu.
FAQs on CBSE Class 10 Mathematics NCERT Solutions Chapter 10 Circles – Free PDF Download 2025-26
1. What is a tangent to a circle in Class 10 Maths Chapter 10?
A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of contact.
Key Facts:
- The tangent is always perpendicular to the radius at the point of contact.
- It never crosses the circle; it only touches it at one place.
- Understanding tangents is crucial for solving questions in NCERT Solutions for Class 10 Maths Chapter 10 Exercise 10.1.
2. How do I prove that tangents drawn from an external point to a circle are equal?
Tangents drawn from an external point to a circle are always equal in length.
Proof steps:
- Let P be a point outside the circle and PA & PB be tangents from P to the points A and B on the circle.
- Join OA and OB (O = centre), also join OP.
- OA = OB = radius; OP is common.
- Triangles OAP and OBP are congruent by RHS (Right angle, Hypotenuse, Side).
- Therefore, PA = PB (by CPCTC).
3. What is the difference between a chord and a tangent of a circle?
The main difference lies in how many points the line shares with the circle.
Key Comparison:
- Chord: A line segment with both endpoints on the circle (e.g., AB).
- Tangent: A line that touches the circle at exactly one point only (point of contact). It never crosses the circle.
4. Which theorems are included in NCERT Class 10 Circles Exercise 10.1?
NCERT Exercise 10.1 covers fundamental theorems important for CBSE board exams:
- Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
- Theorem 2: The lengths of tangents drawn from an external point to a circle are equal.
- Definition and properties of tangent, chord, and related terms.
5. What is the easiest way to prove tangent properties in Class 10 Circles?
Start with a labelled diagram and use congruent triangles:
- Draw clear diagrams marking radii, tangents, and points of contact.
- State given and to prove using NCERT terminology.
- Apply RHS congruence for triangles formed by radius and tangents.
- Conclude with a statement on equality or perpendicularity, as required.
6. Can I download solved PDF solutions for Circles Exercise 10.1 for free?
Yes, you can free download NCERT Solutions for Class 10 Maths Chapter 10 Exercise 10.1 (Circles) in PDF format from trusted educational platforms.
Benefits:
- Offline access for revision
- Exam-ready, stepwise proofs following the latest CBSE 2025 syllabus
- Board-format answer writing tips and practice questions
7. Why are tangents to a circle always perpendicular to the radius?
A tangent to a circle is always perpendicular to the radius at the point of contact due to the circle's geometric property.
This means:
- If a line L touches a circle at point P and O is the centre, then OP ⟂ L.
- This property is frequently used in board exam problems and is proven in NCERT Chapter 10.
8. What common mistakes should I avoid in Circles proofs for CBSE board exams?
To score full marks in Circles proofs, avoid these errors:
- Not drawing a clear, labelled diagram
- Missing important proof steps or statements (e.g., writing 'by CPCTC' for congruency)
- Forgetting to mention units or justify equal tangents
- Skipping the reason for perpendicularity (must write 'Radius is perpendicular to tangent')
- Ignoring NCERT terminology and format
9. How do I write a perfect board answer for an NCERT Circles theorem?
Write your answer stepwise and as per CBSE pattern:
- Start with 'Given', 'To Prove', and 'Construction' (if needed).
- Include a neat, labelled diagram showing all required points and lines.
- Write each proof step in order, justifying with triangle properties or theorems.
- End with a clear concluding statement, such as 'Hence, proved.'
10. How are Vedantu’s NCERT Solutions different from other online answers?
Vedantu’s NCERT Solutions for Class 10 Circles stand out for:
- Stepwise, board-exam-ready answers using latest CBSE 2025 format
- Diagrams and visual explanations for every theorem
- Solutions reviewed by experienced CBSE teachers
- Revision tips, common mistake alerts, and exam hacks included
- Free downloadable PDFs for last-minute preparation and offline study
