Question

In the given figure, if AB || CD || FE then Find x and AE.

Answer 1

Construction: Join AFintersecting CD at X.

In △ABF, DX || AB
$\frac{\mathrm{FD}}{\mathrm{DB}}=\frac{\mathrm{FX}}{\mathrm{XA}}...\left(1\right)\left(\mathrm{By}\mathrm{Basic}\mathrm{proportionality}\mathrm{theorem}\right)$
In △AEF, XC||FE
$\frac{\mathrm{FX}}{\mathrm{XA}}=\frac{\mathrm{EC}}{\mathrm{CA}}...\left(2\right)\left(\mathrm{By}\mathrm{Basic}\mathrm{proportionality}\mathrm{theorem}\right)$
From (1) and (2), we get
$$\begin{gathered} \frac{\mathrm{FD}}{\mathrm{DB}}=\frac{\mathrm{EC}}{\mathrm{CA}} \\ \Rightarrow \frac{4}{8}=\frac{x}{12} \\ \Rightarrow x=6 \end{gathered}$$
Now, AE = AC + CE
= 12 + 6
= 18