Draw a circle with centre O of radius 3.4 cm. Draw a chord MN of length 5.7 cm in it. Construct tangents at point M and N to the circle.

Answer: The tangents at M and N meet at a point T, and TM = TN.

Rough figure: a circle with centre O; chord MN; radii OM and ON; lines through M and N perpendicular to OM and ON meeting at T.

Steps of construction:

1. Draw a circle with centre O and radius 3.4 cm.
2. Take a point M on the circle. With M as centre and radius 5.7 cm, draw an arc cutting the circle at N. Draw chord MN (MN = 5.7 cm).
3. Draw ray OM. At M draw line \(l \perp\) ray OM (by drawing arcs on ray OM on both sides of M and bisecting). Line \(l\) is the tangent at M.
4. Draw ray ON. At N draw line \(m \perp\) ray ON in the same way. Line \(m\) is the tangent at N.
5. Lines \(l\) and \(m\) meet at T. Lines \(l\) and \(m\) are the required tangents.

Final figure (as drawn): circle with centre O (radius 3.4 cm); chord MN = 5.7 cm; rays OM and ON extended beyond the circle; the perpendicular lines at M and N with right-angle marks and their construction arcs; the tangents meeting at T.

Reason: a line perpendicular to a radius at its outer end is a tangent. Since tangent segments from an external point are equal, TM = TN.

(Note: a picture cannot be shown here; in the examination the figure must actually be drawn with ruler and compass, labelled as described, and the construction arcs must be kept.)