\(\Delta XYZ \sim \Delta XLM\). In \(\Delta XYZ\), XY = 4.8 cm, YZ = 5.6 cm and XZ = 4.2 cm, and \(\dfrac{XY}{XL} = \dfrac{4}{3}\). Construct \(\Delta XYZ\) and \(\Delta XLM\).

Answer: \(\Delta XLM\) has XL = 3.6 cm, LM = 4.2 cm, XM = 3.15 cm.

Analysis: \(\dfrac{XY}{XL} = \dfrac{YZ}{LM} = \dfrac{XZ}{XM} = \dfrac{4}{3}\). \(XL = \dfrac{3 \times 4.8}{4} = 3.6\) cm, \(LM = \dfrac{3 \times 5.6}{4} = 4.2\) cm, \(XM = \dfrac{3 \times 4.2}{4} = 3.15\) cm. The two triangles share vertex X.

Steps of construction:

1. Draw seg XY = 4.8 cm. With X as centre radius 4.2 cm and Y as centre radius 5.6 cm draw arcs meeting at Z. Join XZ and YZ.
2. Draw ray XT making an acute angle with XY on the side opposite Z; mark 4 equal parts \(X_1, X_2, X_3, X_4\).
3. Join \(X_4Y\). Through \(X_3\) draw a line parallel to \(X_4Y\), meeting XY at L.
4. Through L draw a line parallel to YZ, meeting XZ at M. \(\Delta XLM\) is the required triangle.

The ray method avoids measuring 3.15 cm. Check: XL ≈ 3.6 cm and LM ≈ 4.2 cm.

[Figure: \(\Delta XYZ\) with the smaller \(\Delta XLM\) inside it at vertex X, LM parallel to YZ.]