Class 10 Maths Chapter 9 Exercise 9.1 Solutions 2026 – Tangent and Angles of a Circle Notes PDF

Unit 9: Tangent and Angles of a Circle | Exercise 9.1 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 9.1 solutions for Tangent and Angles of a Circle have been uploaded below and are free to view or download, fully solved according to the new PCTB/PECTAA Class 10 Mathematics syllabus.

A Line That Touches, Never Crosses

A tangent is a special kind of line that touches a circle at exactly one point without ever cutting through it — quite different from a chord, which cuts across the interior. Exercise 9.1 introduces tangents formally, along with the single most important property they all share: a tangent is always perpendicular to the radius drawn to the point of contact.

Key Concepts Covered in Exercise 9.1

Definition of a Tangent

A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of contact or point of tangency. Unlike a secant, which intersects the circle at two points, a tangent never enters the circle’s interior.

The Tangent-Radius Perpendicularity Theorem

At the point of contact, a tangent is always perpendicular to the radius drawn to that point. This single fact is the foundation for nearly every calculation involving tangents, since it guarantees a right angle to work with.

Tangents from an External Point

From any point outside a circle, exactly two tangents can be drawn to the circle, and these two tangent lengths (measured from the external point to each point of contact) are always equal.

Using the Pythagorean Theorem with Tangents

Because the tangent-radius perpendicularity theorem guarantees a right angle, the Pythagorean theorem is frequently used to relate the tangent length, the radius, and the distance from the external point to the center.

Step-by-Step Solved Examples

Q. No. 1: A tangent is drawn from an external point 13 cm from the center of a circle of radius 5 cm. Find the length of the tangent.

The radius, tangent, and distance to center form a right triangle (right angle at the point of contact).

tangent² = distance² − radius² = 13² − 5² = 169 − 25 = 144

tangent = 12 cm

Q. No. 2: Two tangents PA and PB are drawn from an external point P to a circle. If PA = 8 cm, find PB.

Since tangents from the same external point are always equal in length:

PB = PA = 8 cm

Q. No. 3: A circle has radius 6 cm, and a tangent of length 8 cm is drawn from an external point P. Find the distance of P from the center.

distance² = tangent² + radius² = 8² + 6² = 64 + 36 = 100

distance = 10 cm

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 9.1 in the same numbered, step-by-step format.

MCQs, Short Questions & Long Questions from This Exercise

Alongside the main exercise, this page’s uploaded notes are organized the way Punjab Board papers are structured, so students can practice by question type:

• MCQs: Quick questions on the number of tangents that can be drawn from a given point
• Short Questions: Finding a tangent length using the Pythagorean theorem, like Q. No. 1 above
• Long Questions: Multi-step problems combining equal tangents with other circle theorems

Where This Matters Beyond This Exercise

The perpendicularity and equal-tangent theorems established here are used directly in the next exercise, which explores angles formed between tangents, chords, and secants. Tangent lines also appear throughout engineering and physics — describing the instantaneous direction of motion along a curved path, for instance.

Common Mistakes Students Make in Exercise 9.1

• Forgetting that the right angle in a tangent problem is always at the point of contact, not at the external point
• Mixing up which side of the right triangle is the tangent, the radius, and the distance to the center
• Assuming a tangent and a secant behave the same way, when a secant passes through two points on the circle
• Forgetting that exactly two (not one, and not infinitely many) tangents exist from an external point

Why This Exercise Matters for the Board Exam

Tangent-length questions using the Pythagorean theorem are short, formulaic, and one of the most reliably repeated question types in this chapter, making Exercise 9.1 an efficient source of marks. A clearly labeled diagram showing the right angle at the point of contact is often specifically rewarded with its own mark.

Quick Links – Chapter 9: Tangent and Angles of a Circle

Links for sections other than Exercise 9.1 will be activated as they are published on this site.

Download Exercise 9.1 Notes PDF

Click below to view or download the complete Exercise 9.1 notes in PDF format.

[ Download PDF ]      [ View Online ]

(Google Drive / download links to be embedded here.)

Explore More Class 10 Notes

• Class 10 Maths Chapter 6 – Vectors in Plane (All Exercises)
• Class 10 Maths Chapter 7 – Trigonometry (All Exercises)
• Class 10 Maths Chapter 8 – Chords and Arcs of a Circle (All Exercises)
• Class 10 Maths Chapter 9 – Full Chapter Notes (All Exercises)
• Class 10 English Notes
• Class 10 Physics Notes
• Class 10 Computer Science Notes

Join Our Community

Stay updated with new notes, guess papers, and past papers by following us:

• Facebook Page — join thousands of students
• YouTube Channel — video lectures and explanations
• WhatsApp Group — instant updates and doubt-solving

Frequently Asked Questions (FAQs)

Q1. Are these Exercise 9.1 notes free to download?

Yes, all notes on this page are completely free to view and download in PDF format.

Q2. Which board are these notes for?

These notes are prepared according to the Punjab Board syllabus and are useful for all Punjab boards (Lahore, Gujranwala, Multan, Sargodha, Rawalpindi, Faisalabad, DG Khan, Bahawalpur, Sahiwal) as well as the Federal Board (FBISE).

Q3. How many tangents can be drawn to a circle from a given point?

From a point on the circle, exactly one tangent can be drawn. From a point outside the circle, exactly two tangents can be drawn, and they are always equal in length. From a point inside the circle, no tangent can be drawn at all.

Q4. Does this page include MCQs and short questions too?

Yes, the uploaded notes include MCQs, short questions, and long questions related to this exercise’s topics, organized by question type to match the board paper pattern.

Q5. How can I download the PDF?

Click the “Download PDF” button above and the notes will open or download directly to your device.

Q6. Are these notes updated for the new 2026–27 syllabus?

Yes, these notes are prepared strictly according to the latest PCTB/PECTAA syllabus for the 2026–27 academic session.

Comments & Feedback

Have a question about any part of Exercise 9.1 or found something missing? Share your comments and questions below — we’d love to help!