Class 10 Maths Chapter 8 Exercise 8.1 Solutions 2026 – Chords and Arcs of a Circle Notes PDF

Unit 8: Chords and Arcs of a Circle | Exercise 8.1 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 8.1 solutions for Chords and Arcs 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.

The Hidden Symmetry Inside Every Circle

A chord is simply a straight line joining two points on a circle, but the way chords relate to a circle’s center follows several precise, provable rules. Exercise 8.1 introduces the core chord theorems — relationships that hold for every circle, regardless of its size — and gives practice applying them to find unknown lengths and angles.

Key Concepts Covered in Exercise 8.1

The Perpendicular Bisector Theorem

A line drawn from the center of a circle perpendicular to a chord always bisects that chord — splitting it into two exactly equal halves. The converse is also true: a line from the center that bisects a chord must be perpendicular to it.

Equal Chords and Equal Angles

Equal chords in the same circle (or in congruent circles) always subtend equal angles at the center. Conversely, if two chords subtend equal angles at the center, the chords themselves must be equal in length.

Equal Chords and Distance from the Center

Equal chords in the same circle are always equidistant from the center — meaning the perpendicular distance from the center to each chord is the same. Conversely, chords equidistant from the center are equal in length.

Using the Radius in Chord Problems

Since the perpendicular from the center to a chord creates a right triangle (with the radius as hypotenuse, half the chord as one leg, and the perpendicular distance as the other), the Pythagorean theorem is frequently used alongside these chord theorems to find missing lengths.

Step-by-Step Solved Examples

Q. No. 1: A chord of length 16 cm is drawn in a circle of radius 10 cm. Find the distance of the chord from the center.

Half the chord = 16/2 = 8 cm

Using the Pythagorean theorem: radius² = (half chord)² + distance²

10² = 8² + d²  →  100 = 64 + d²  →  d² = 36

d = 6 cm

Q. No. 2: Two equal chords of a circle subtend angles of 70° and (x+10)° at the center. Find x.

Since the chords are equal, the angles they subtend at the center must also be equal:

70 = x + 10

x = 60

Q. No. 3: In a circle of radius 13 cm, a chord is 24 cm long. Find its distance from the center, and verify using the Pythagorean theorem.

Half the chord = 24/2 = 12 cm

d² = radius² − (half chord)² = 13² − 12² = 169 − 144 = 25

d = 5 cm

Verify: 12² + 5² = 144 + 25 = 169 = 13² ✔ confirmed

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 8.1 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 applying the perpendicular bisector theorem or the equal-chords rule
• Short Questions: Finding a chord’s distance from the center given the radius, like Q. No. 1 above
• Long Questions: Multi-step problems combining two chord theorems, such as Q. No. 2 or 3 above

Where This Matters Beyond This Exercise

These chord theorems form the geometric foundation for arc-angle relationships covered in the next exercise, and for tangent properties studied in the chapter after this one. Circle geometry theorems like these also appear frequently in engineering and design, wherever precise, provable relationships between curves and straight lines are needed.

Common Mistakes Students Make in Exercise 8.1

• Forgetting to halve the chord length before applying the Pythagorean theorem
• Assuming any two chords that look similar in a diagram are automatically equal, without proof
• Confusing the distance from the center to a chord with the chord’s own length
• Skipping the verification step in Pythagorean-theorem-based problems, missing arithmetic errors

Why This Exercise Matters for the Board Exam

Chord-distance problems combining the perpendicular bisector theorem with the Pythagorean theorem are a favorite, reliable long-question style in this chapter, since they test both a geometric theorem and a calculation together. A clearly labeled diagram showing the perpendicular, the radius, and half the chord is often specifically rewarded with its own mark.

Quick Links – Chapter 8: Chords and Arcs of a Circle

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Download Exercise 8.1 Notes PDF

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 8.1 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. Do I need to know earlier chapters before starting Exercise 8.1?

Basic geometry and the Pythagorean theorem from earlier classes are used throughout this exercise, but no prior trigonometry or algebra chapters from this book are required.

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

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