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

Steps of Construction:

1. Draw a line segment and mark centre O on it. Draw a circle of radius 3 cm with centre O.
2. Mark points P and Q on the extended diameter, each at 7 cm from O, on opposite sides.
3. Find the midpoint M₁ of OP. Draw a circle with centre M₁ and radius M₁O. Let it intersect the given circle at points A and B.
4. Join PA and PB. These are the required tangents from P.
5. Similarly, find midpoint M₂ of OQ. Draw a circle with centre M₂ and radius M₂O. Let it intersect the given circle at points C and D.
6. Join QC and QD. These are the required tangents from Q.

Justification: Since OP is the diameter of the auxiliary circle, ∠OAP = 90° (angle in semicircle), so PA ⊥ OA, making PA a tangent. Similarly for all other tangents.

Source: Chapter 10, Constructions (Tangents from an External Point)

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• This is a construction question; examiners award marks for correct steps, not just the figure.
• Key idea: To draw tangents from external point P, draw a circle with OP as diameter — it intersects the given circle at the points of contact (since the angle in a semicircle = 90°, confirming the radius–tangent perpendicularity from Theorem 10.1).
• Since P and Q are on opposite sides of centre O (each 7 cm away), you get two tangents from each point (4 tangents total).
• Always include a brief justification — even in 3-mark construction questions, 1 mark is typically for reasoning.