In triangle ABC, M is the mid-point of AB and a straight line through M and parallel to BC cuts AC at N. Find the lengths of AN and MN, if BC = 7 cm and AC = 5 cm.

By converse of mid-point theorem,

The straight line drawn through the mid-point of one side of a triangle parallel to another, bisects the third side.

∴ N bisects AC.

∴ AN = $\dfrac{AC}{2} = \dfrac{5}{2}$ = 2.5 cm.

By mid-point theorem,

The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is equal to half of it.

∴ MN = $\dfrac{1}{2}BC = \dfrac{1}{2} \times 7$ = 3.5 cm.

Hence, AN = 2.5 cm and MN = 3.5 cm.