Question

In the fig 2.44, $\square$LMNP is a trapezium in which seg PN $\parallel$seg LM. If PN = 12, LA = 5, LM = 8, show that
(i) $△$LAM $~$$△$NAP
(ii) Find AN

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{In}△\mathrm{LAM}\mathrm{and}△\mathrm{NAP},\mathrm{we}\mathrm{have}: \\ \angle \mathrm{MLN}=\angle \mathrm{LNP}(\mathrm{Alternate}\mathrm{angles};\mathrm{PN}\parallel \mathrm{LM}) \\ \angle \mathrm{LMP}=\angle \mathrm{MPN}(\mathrm{Alternate}\mathrm{angles};\mathrm{PN}\parallel \mathrm{LM}) \\ \therefore △\mathrm{LAM}\cong △\mathrm{NAP}(\mathrm{AA}\mathrm{similarity}\mathrm{test}) \\ \left(\mathrm{ii}\right)△\mathrm{LAM}\cong △\mathrm{NAP},\mathrm{we}\mathrm{have}: \\ \frac{\mathrm{LA}}{\mathrm{NA}}=\frac{\mathrm{LM}}{\mathrm{NP}}=\frac{\mathrm{AM}}{\mathrm{AP}} \\ \Rightarrow \frac{5}{\mathrm{AN}}=\frac{8}{12} \\ \Rightarrow \mathrm{AN}=7.5 \end{gathered}$$