Đáp án:

$\begin{array}{l}
1)a)Xet:\Delta ABC:KE//BC\\
Theo\,Talet:\\
\dfrac{{AK}}{{KB}} = \dfrac{{AE}}{{EC}}\\
 \Leftrightarrow \dfrac{x}{{18}} = \dfrac{5}{{15}} = \dfrac{1}{3}\\
 \Leftrightarrow x = \dfrac{1}{3}.18 = 6\\
Vậy\,x = 6\\
b)Trong:\Delta ABC \bot A\\
Theo\,Pytago:\\
A{B^2} + A{C^2} = B{C^2}\\
 \Leftrightarrow B{C^2} = {6^2} + {8^2} = 100\\
 \Leftrightarrow BC = 10\left( {cm} \right)\\
AB \bot AC;NM \bot AC\\
 \Leftrightarrow AB//NM\\
Theo\,Talet:\\
\dfrac{{CN}}{{BC}} = \dfrac{{CM}}{{AC}}\\
 \Leftrightarrow \dfrac{x}{{10}} = \dfrac{2}{8} = \dfrac{1}{4}\\
 \Leftrightarrow x = \dfrac{1}{4}.10 = 2,5\left( {cm} \right)\\
Vậy\,x = 2,5cm\\
B2)\\
Trong:\Delta ABC:DE//BC\\
Theo\,Talet:\\
\dfrac{{AD}}{{AB}} = \dfrac{{AE}}{{AC}}\\
 \Leftrightarrow \dfrac{{AE}}{{AC}} = \dfrac{6}{9} = \dfrac{2}{3}\\
Trong:\Delta ADC:EF//CD\\
Theo\,Talet:\\
\dfrac{{AE}}{{AC}} = \dfrac{{AF}}{{AD}} = \dfrac{2}{3}\\
 \Leftrightarrow AF = \dfrac{2}{3}.AD = \dfrac{2}{3}.6 = 4\left( {cm} \right)\\
Vậy\,AF = 4cm
\end{array}$