Asked in: March 2025 board paper (the same type was asked in July 2024 and March 2026)Draw a circle with radius 4.1 cm. Construct tangents to the circle from a point at a distance 7.3 cm from the centre.

Answer: Two tangents PA and PB, each about 6.0 cm long.

Steps of construction:

1. Draw a circle with centre O and radius 4.1 cm.
2. Take point P such that OP = 7.3 cm and draw seg OP.
3. Draw the perpendicular bisector of seg OP; let it cut OP at M, the midpoint of OP.
4. With M as centre and radius MO (= 3.65 cm), draw a circle. It cuts the given circle at points A and B.
5. Draw lines PA and PB. These are the required tangents.

[Figure: circle with centre O, radius 4.1 cm; external point P, OP = 7.3 cm; auxiliary circle on diameter OP with centre M; tangents PA and PB touching at A and B.]

Why it works: A and B lie on the circle with diameter OP, so \(\angle OAP = \angle OBP = 90^\circ\) (angle in a semicircle). A line perpendicular to a radius at its outer end is a tangent.

Check: each tangent segment should measure \(\sqrt{7.3^2 - 4.1^2} \approx 6.0\) cm, and PA = PB.