Class 10 Maths Chapter 9 Review Exercise Solutions 2026 – Tangent and Angles of a Circle Notes PDF
Unit 9: Tangent and Angles of a Circle | Review Exercise | Punjab Board New Syllabus 2026–27
Updated August 2026: Review Exercise 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.
Testing the Whole Chapter at Once
Exercise 9.1 focused on tangent properties, and Exercise 9.2 turned to angle theorems. The Review Exercise removes those labels and mixes both sets of theorems together, the way a real board paper naturally combines tangent and angle reasoning inside a single question.
What This Review Exercise Draws On
Tangent Properties
Quick-recall and applied questions on the tangent-radius perpendicularity theorem and equal tangents from an external point, exactly as introduced in Exercise 9.1.
Angle Theorems
Mixed problems applying the central angle theorem, the angle-in-a-semicircle result, and the angles-in-the-same-segment theorem, as covered in Exercise 9.2.
Combining Tangent and Angle Reasoning
Some questions in this review require recognizing that a tangent creates a right angle at the point of contact, which can then be used alongside the chapter’s angle theorems within a single larger figure.
Step-by-Step Solved Examples
Q. No. 1: A tangent of length 15 cm is drawn from a point 17 cm from the center of a circle. Find the radius, then find the angle subtended by the diameter at any point on the circle.
radius² = distance² − tangent² = 17² − 15² = 289 − 225 = 64
radius = 8 cm
The diameter always subtends a 90° angle at any point on the circle (angle in a semicircle).
Q. No. 2: Two tangents PA and PB are drawn from an external point P, with ∠APB = 50°. If OA and OB are radii to the points of contact, find ∠AOB (where O is the center).
Since PA ⊥ OA and PB ⊥ OB (tangent-radius perpendicularity), quadrilateral OAPB has two 90° angles.
Angle sum of quadrilateral OAPB = 360°
∠AOB + ∠APB + 90° + 90° = 360°
∠AOB = 360° − 50° − 180° = 130°
Q. No. 3: An arc subtends an angle of 96° at the center. Points C and D lie on the major arc’s remaining segment. Find ∠ACB and ∠ADB.
Angle at circumference = (1/2) × central angle = (1/2) × 96° = 48°
Since C and D are both in the same segment, subtending the same arc:
∠ACB = ∠ADB = 48° (angles in the same segment are equal)
These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of the Review Exercise in the same numbered, step-by-step format.
MCQs, Short Questions & Long Questions from This Review
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: Rapid-fire recall across both exercises, useful for timed practice
- Short Questions: Two-step problems combining any two of the chapter’s core ideas
- Long Questions: Full multi-part problems that mirror the structure of an actual board exam question
Study Tips for Reviewing This Chapter
- Attempt the Review Exercise only after completing Exercises 9.1 and 9.2
- Keep the tangent and angle theorems written together on one page — many long questions combine both in the same figure
- Practice sketching a quick diagram for every question, even short ones, to correctly place right angles and equal angles
- Re-attempt any question you got wrong the next day, rather than only reading the correct answer once
Common Mistakes Students Make in This Review Exercise
- Trying to answer a multi-step question in one line instead of solving it stage by stage
- Applying a tangent theorem where an angle theorem is needed, or vice versa
- Forgetting that a quadrilateral formed by two tangents and two radii has two guaranteed 90° angles
Why This Exercise Matters for the Board Exam
Board papers rarely test tangent or angle theorems in complete isolation, which is exactly what this Review Exercise simulates. Working through it carefully is one of the best ways to confirm genuine readiness for Chapter 9 before moving on to Chapter 10.
Quick Links – Chapter 9: Tangent and Angles of a Circle
| Section | Covers |
| Exercise 9.1 | Tangent properties |
| Exercise 9.2 | Angles in a circle |
| Short Questions | Single-concept practice |
| Chapter 9 MCQs | Objective-type practice |
| Review Exercise | You are here |
Links for sections other than the Review Exercise will be activated as they are published on this site.
Download Review Exercise Notes PDF
Click below to view or download the complete Review Exercise notes in PDF format.
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Frequently Asked Questions (FAQs)
Q1. Are these Review Exercise 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. When should I attempt the Review Exercise?
Only after completing both exercises (9.1 and 9.2), since the questions here mix tangent and angle topics without labeling which exercise each one came from.
Q4. Does this page include MCQs and short questions too?
Yes, the uploaded notes include MCQs, short questions, and long questions covering the whole chapter, 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
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