Draw a circle of radius 3.4 cm and centre E. Take a point F on the circle. Take another point A such that E-F-A and FA = 4.1 cm. Draw tangents to the circle from point A.

Answer: EA = 3.4 + 4.1 = 7.5 cm. The circle with diameter EA cuts the given circle at P and Q; AP and AQ are the tangents (each ≈ 6.7 cm).

Steps of construction:

1. Draw a circle with centre E and radius 3.4 cm. Take point F on it.
2. Draw ray EF and mark A on it beyond F with FA = 4.1 cm, so E-F-A and EA = 7.5 cm.
3. Draw the perpendicular bisector of seg EA to get its midpoint M.
4. With centre M and radius ME draw a circle; it cuts the given circle at P and Q.
5. Draw lines AP and AQ. These are the required tangents.

Reason: \(\angle EPA = \angle EQA = 90^\circ\) (angles in a semicircle), so AP and AQ are perpendicular to radii at their outer ends. Each tangent segment \(= \sqrt{7.5^2-3.4^2}\approx 6.7\) cm.