Draw a circle of radius $3$ cm. Take two points P and Q on one of its extended diameter each at a distance of $7$ cm from its centre. Construct tangents to the circle from these two points P and Q.
Generated by claude-sonnet-4-6 · 2026-06-15 10:24 · grounding rag
Model Answer
Steps of Construction:
- Draw a line segment and mark centre O on it. Draw a circle of radius 3 cm with centre O.
- Mark points P and Q on the extended diameter, each at 7 cm from O, on opposite sides.
- 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.
- Join PA and PB. These are the required tangents from P.
- 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.
- 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)
---
Explanation
- 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.