Asked in: March 2026 board paperDraw a circle with centre O and radius 3.5 cm, take a point P at a distance 5.7 cm from the centre. Draw tangents to the circle from point P.

Answer: Two tangents PA and PB, each about 4.5 cm long.

Steps of construction:

1. Draw a circle with centre O and radius 3.5 cm.
2. Take point P such that OP = 5.7 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 (= 2.85 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 3.5 cm; external point P, OP = 5.7 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 \(\sqrt{5.7^2 - 3.5^2} \approx 4.5\) cm, and PA = PB.