Draw a circle of radius 4 cm. From a point P, 9 cm from the centre, draw the pair of tangents and measure their lengths. Verify by calculation.

1. 1 · Write what is given and what is to be constructed Radius r = 4 cm and OP = 9 cm. Draw the tangents PA and PB, measure them and verify.
2. 2 · Draw the rough figure Circle with centre O, point P outside, midpoint M of OP, the helper circle, and tangents PA and PB.
3. 3 · Write the theorem it uses The angle in a semicircle is 90°. A line perpendicular to the radius at its end point is a tangent.
4. 4 · Construct
   • Circle with centre O and radius 4 cm; mark P with OP = 9 cm
   • Perpendicular bisector of OP gives the midpoint M
   • Circle with centre M and radius MO = 4.5 cm cuts the circle at A and B
   • Join PA and PB
5. 5 · Write the steps of construction Number each step in short sentences, in the order you drew them.
6. 6 · Measure and verify Measured PA = PB ≈ 8.1 cm. By calculation, PA = √(9² − 4²) = √65 ≈ 8.06 cm.
7. 7 · Check and box the answer Box: PA and PB are the required tangents, each about 8.1 cm long.

Scoring tip: Write the verification line. Examiners give marks for checking the measured length with √(OP² − r²).