IMC 2020 Q15

The diagrams show one of the twenty-five congruent right-angled triangles and the two such triangles at vertex R.
Let the two acute angles in each of the triangles be \(x°\) and \(y°\).

Note that \(x + y + 90 = 180\), as the interior angles of a triangle sum to 180°.
So \(x + y = 90\). In the second diagram, the two angles which meet at R are \(x°\) and \(y°\), so we can deduce that angle \(QRP\) is a right angle.
Let the length of the hypotenuse of each small triangle be \(a\) cm.
Note that angle \(QRP\) is a right angle, \(PR\) has length \(4a\) cm and \(RQ\) has length \(3a\) cm.
So the lengths of the sides in triangle \(PQR\) are in the ratio 3 : 4 : 5.
Therefore the length of \(PQ\), in cm, is \(5 \times \tfrac{2.4}{4} = 3\).