In a quadrilateral ABCD the diagonals AC and BD measure 18 cm and 24 cm. The points P, Q, R and S are the midpoints of the sides AB, BC, CD and DA respectively. What is the perimeter of quadrilateral PQRS?

PQ joins the midpoints of AB and BC, so by the mid-point theorem PQ = ½ × AC = 9 cm, and RS (in triangle ACD) is also 9 cm. QR and SP are the mid-segments of triangles BCD and ABD, so each is ½ × BD = 12 cm. The perimeter is 9 + 12 + 9 + 12 = 42 cm — always the SUM of the two diagonals. Halving only once gives the trap 21 cm, not halving at all gives 84 cm, and 24 cm is merely the longer diagonal.