Mathematics — Standard (041) — CBSE Class 10 board question

Steps of Construction:

1. Draw two concentric circles with the same centre O, of radii 2 cm and 5 cm.
2. Take any point P on the outer circle (radius 5 cm).
3. Join OP and find its midpoint M (bisect OP).
4. Draw a circle with centre M and radius MO (= MP). This circle intersects the inner circle at points T₁ and T₂.
5. Join PT₁ and PT₂.

PT₁ and PT₂ are the required tangents to the inner circle from point P.

Justification: ∠OT₁P = 90° (angle in a semicircle), so OT₁ ⊥ PT₁, meaning PT₁ is tangent to the inner circle. Similarly for PT₂.

Source: Chapter 10, Circles

---

• The key idea is that the tangent is perpendicular to the radius at the point of contact (Theorem 10.1), so ∠OT₁P = 90°.
• To find T₁ and T₂, draw a circle on OP as diameter — any point on this circle sees OP at 90°, giving the tangent points where it cuts the inner circle.
• Examiners award marks for: correct diagram, clear steps, and a one-line justification. Skipping justification costs 1 mark.