Draw a circle with centre P and radius 3.4 cm. Take point Q at a distance 5.5 cm from the centre. Construct tangents to the circle from point Q.

Answer: Draw the circle with diameter PQ; it cuts the given circle at A and B. QA and QB are the tangents (each ≈ 4.3 cm).

Steps of construction:

1. Draw a circle with centre P and radius 3.4 cm.
2. Mark point Q such that PQ = 5.5 cm and draw seg PQ.
3. Draw the perpendicular bisector of seg PQ; let it cut PQ at M (the midpoint).
4. With centre M and radius MP, draw a circle. Name the points where it cuts the given circle as A and B.
5. Draw lines QA and QB. These are the required tangents.

Why it works: \(\angle PAQ\) and \(\angle PBQ\) are angles inscribed in a semicircle (diameter PQ), so each is \(90^\circ\). A line perpendicular to a radius at its outer end is a tangent.

Check: each tangent segment \(= \sqrt{5.5^2 - 3.4^2} \approx 4.3\) cm, and the two tangent segments are equal.