Construct ΔPYQ such that, PY = 6.3 cm, YQ = 7.2 cm, PQ = 5.8 cm. If YZ/YQ = 6/5, then construct ΔXYZ similar to ΔPYQ.

Answer: ΔXYZ has XY = 7.56 cm, YZ = 8.64 cm, XZ = 6.96 cm.

Analysis: ΔXYZ ~ ΔPYQ (X↔P, Y↔Y, Z↔Q), so \(\dfrac{XY}{PY}=\dfrac{YZ}{YQ}=\dfrac{XZ}{PQ}=\dfrac65\). ΔXYZ is bigger and Y is common, so Z lies on ray YQ and X on ray YP.

\(YZ = 7.2\times\tfrac65 = 8.64\), \(XY = 6.3\times\tfrac65 = 7.56\), \(XZ = 5.8\times\tfrac65 = 6.96\) (cm).

Steps of construction:

1. Draw seg YQ = 7.2 cm. With centre Y radius 6.3 cm and centre Q radius 5.8 cm draw arcs meeting at P. Join PY, PQ.
2. Draw ray YT making an acute angle with YQ, on the side opposite to P. Mark 6 equal parts \(Y_1,\dots,Y_6\).
3. Join \(Y_5Q\). Through \(Y_6\) draw a line parallel to \(Y_5Q\), meeting ray YQ (extended) at Z.
4. Through Z draw a line parallel to QP, meeting ray YP (extended) at X.
5. ΔXYZ is the required triangle.