Class 10 Maths Chapter 10 Exercise 10.2 Solutions 2026 – Practical Geometry of Circles Notes PDF

Unit 10: Practical Geometry of Circles | Exercise 10.2 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 10.2 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.

Constructing the Tangent Itself

Exercise 9.1 proved that a tangent is always perpendicular to the radius at the point of contact — Exercise 10.2 turns that theorem into an actual ruler-and-compass construction, drawing tangent lines to a circle both from a point on the circle and from a point outside it.

Key Concepts Covered in Exercise 10.2

Tangent at a Point on the Circle

To construct a tangent at a given point P on a circle, join P to the center O, then construct a line through P perpendicular to OP. This works directly from the tangent-radius perpendicularity theorem.

Tangents from an External Point

To construct both tangents from an external point P to a circle with center O: join OP, bisect OP to find its midpoint M, then draw a circle centered at M with radius MO (or MP). This new circle intersects the original circle at exactly the two points of tangency.

Why the Midpoint Construction Works

The circle centered at M passes through both O and P, and any angle inscribed in a semicircle is 90° (from Exercise 9.2) — so the angle at each intersection point, OTP, is automatically a right angle, exactly satisfying the tangent-radius perpendicularity condition.

Verifying the Construction

After construction, the tangent length can be checked using the Pythagorean theorem from Exercise 9.1: tangent² = OP² − radius², comparing the measured length to the calculated one.

Step-by-Step Solved Examples

Q. No. 1: Describe the construction steps to draw a tangent at point P on a circle with center O.

1. Draw the radius OP and extend it slightly beyond P.

2. At point P, construct a line perpendicular to OP (using a compass to mark equal arcs and bisect the angle, or using a set square).

3. This perpendicular line is the required tangent at P.

Q. No. 2: Describe the construction steps to draw two tangents from an external point P to a circle with center O and radius 4 cm, where OP = 9 cm.

1. Draw the circle with center O and radius 4 cm, and mark external point P with OP = 9 cm.

2. Join OP and construct its perpendicular bisector to find midpoint M.

3. With center M and radius MO (= MP = 4.5 cm), draw a circle.

4. This circle intersects the original circle at two points, T₁ and T₂ — join P to each for the two tangents.

Q. No. 3: For the construction in Q. No. 2, calculate the expected tangent length to verify the construction.

tangent² = OP² − radius² = 9² − 4² = 81 − 16 = 65

tangent = √65 ≈ 8.06 cm

The measured tangent length in the construction should closely match this value.

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 10.2 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 the first step of constructing a tangent from an external point
• Short Questions: Stating the construction steps for a tangent at a point on the circle
• Long Questions: Full ruler-and-compass construction of tangents from an external point, with length verification

Where This Matters Beyond This Exercise

This construction directly applies the tangent-radius theorem (Chapter 9) and the angle-in-a-semicircle theorem (also Chapter 9), showing how earlier theoretical results become practical drawing techniques. These same tangent-construction principles are used in mechanical engineering and CAD software, wherever a smoothly curved and straight edge need to meet without a sharp corner.

Common Mistakes Students Make in Exercise 10.2

• Forgetting to bisect OP before drawing the second circle, which is essential to locate M correctly
• Using the wrong radius for the second circle — it must be MO (half of OP), not the original circle’s radius
• Constructing only one tangent when the question asks for both
• Skipping the verification step, missing a construction error

Why This Exercise Matters for the Board Exam

Like Exercise 10.1, construction questions here are graded stage by stage, so following the correct sequence — join OP, bisect it, draw the second circle, mark the tangent points — matters as much as the final result. This is one of the few topics where physical practice with real instruments makes a measurable difference in exam performance.

Quick Links – Chapter 10: Practical Geometry of Circles

Links for sections other than Exercise 10.2 will be activated as they are published on this site.

Download Exercise 10.2 Notes PDF

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 10.2 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. Why does bisecting OP help construct the tangents?

The midpoint of OP becomes the center of a new circle that guarantees a 90° angle at each intersection point with the original circle, exactly matching the tangent-radius perpendicularity condition — no additional angle measurement is needed.

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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