Class 10 Maths Chapter 10 Exercise 10.1 Solutions 2026 – Practical Geometry of Circles Notes PDF

Unit 10: Practical Geometry of Circles | Exercise 10.1 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 10.1 solutions for Practical Geometry of Circles have been uploaded below and are free to view or download, fully solved according to the new PCTB/PECTAA Class 10 Mathematics syllabus.

Turning Theorems into Constructions

Every circle theorem studied so far describes a relationship; this chapter asks students to actually draw circles that satisfy those relationships, using only a ruler and compass. Exercise 10.1 focuses on two related constructions: the circle inscribed inside a triangle, and the circle circumscribed around one.

Key Concepts Covered in Exercise 10.1

The Circumscribed Circle (Circumcircle)

The circumcircle of a triangle passes through all three vertices. Its center, called the circumcenter, is found by constructing the perpendicular bisectors of any two sides of the triangle — the point where they intersect is equidistant from all three vertices.

The Inscribed Circle (Incircle)

The incircle of a triangle touches all three sides from the inside. Its center, called the incenter, is found by constructing the angle bisectors of any two angles of the triangle — the point where they intersect is equidistant from all three sides.

Finding the Radius

For the circumcircle, the radius is the distance from the circumcenter to any vertex. For the incircle, the radius is the perpendicular distance from the incenter to any side, since the circle must just touch each side without crossing it.

Construction Steps and Accuracy

Both constructions rely on precise use of a compass to bisect angles or sides. This exercise gives practice following the construction steps in the correct order and checking the result for accuracy, since even a small error compounds by the final circle.

Step-by-Step Solved Examples

Q. No. 1: Describe the construction steps to find the circumcenter of a triangle ABC.

1. Construct the perpendicular bisector of side AB.

2. Construct the perpendicular bisector of side BC.

3. Mark the point where the two perpendicular bisectors intersect — this is the circumcenter O.

4. Measure OA (or OB or OC) as the radius, and draw the circumcircle through all three vertices.

Q. No. 2: Describe the construction steps to find the incenter of a triangle ABC.

1. Construct the angle bisector of angle A.

2. Construct the angle bisector of angle B.

3. Mark the point where the two angle bisectors intersect — this is the incenter I.

4. Drop a perpendicular from I to any one side to find the inradius, then draw the incircle.

Q. No. 3: In an equilateral triangle, explain why the circumcenter and the incenter are the same point.

In an equilateral triangle, every perpendicular bisector of a side also passes through the opposite vertex and bisects that vertex’s angle.

This means the perpendicular bisectors and the angle bisectors are exactly the same three lines.

So their intersection point — the circumcenter and incenter — must coincide at the same point (the centroid).

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 10.1 in the same numbered, step-by-step format, including full diagrams.

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 identifying whether a construction uses perpendicular bisectors or angle bisectors
• Short Questions: Stating the construction steps for the circumcenter or incenter
• Long Questions: Full ruler-and-compass constructions of the circumcircle or incircle for a given triangle

Where This Matters Beyond This Exercise

These constructions rely directly on the chord and angle theorems from Chapters 8 and 9 — the perpendicular bisector of a chord passing through the center is exactly the same idea used to find the circumcenter here. These construction skills also apply directly in engineering drawing and design, wherever a circle needs to be fitted precisely to a given shape.

Common Mistakes Students Make in Exercise 10.1

• Constructing an angle bisector when a perpendicular bisector was needed, or vice versa
• Using an imprecise compass setting, leading to a circle that doesn’t pass through all three vertices exactly
• Forgetting to extend construction lines far enough for them to visibly intersect
• Mixing up the circumcenter (outside the triangle for obtuse triangles) with the incenter (always inside)

Why This Exercise Matters for the Board Exam

Construction questions are graded step-by-step, with marks awarded for each correct stage of the process, not just the final circle — making a clear, methodical approach essential. Since these are practical, hands-on questions, practicing the physical construction with a real ruler and compass is far more effective than only reading through the steps.

Quick Links – Chapter 10: Practical Geometry of Circles

Links for sections other than Exercise 10.1 will be activated as they are published on this site.

Download Exercise 10.1 Notes PDF

Click below to view or download the complete Exercise 10.1 notes in PDF format.

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 10.1 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. What tools do I need for this exercise?

A ruler, a compass, and a protractor (for checking angles) are the standard tools needed for the constructions in this exercise, exactly as required in the board exam’s practical geometry section.

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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