17. In rectangle $J K L M$, the bisector of angle $K J M$ cuts the diagonal $K M$ at point $N$ as shown. The distances between $N$ and sides $L M$ and $K L$ are $8 \mathrm{~cm}$ and $1 \mathrm{~cm}$ respectively. The length of $K L$ is $(a+\sqrt{b}) \mathrm{cm}$. What is the value of $a+b$ ?
17. 16

Let points $P$ and $Q$ be the points where the perpendiculars from $N$ to $M L$ and $K L$ meet the lines and extend line $P N$ so it meets $J K$ at $R$, as shown in the diagram. Since $J N$ is the bisector of angle $M J K$, angle $N J R=45^{\circ}$. Since angle $J R N$ is $90^{\circ}$, triangle $J R N$ is isosceles and $J R=R N$. Let the length of $R N$ be $x \mathrm{~cm}$. Hence the lengths of $J R$ and $P M$ are also $x \mathrm{~cm}$. Observe that triangles $N K Q$ and $M N P$ are similar since they have the same angles. Therefore $\frac{1}{x}=\frac{x}{8}$ and so $x=\sqrt{8}$ since $x$ is positive. The length of $K L$ is equal to the sum of the lengths of $N P$ and $N R$. Therefore, the length of $K L$ is $(8+\sqrt{8}) \mathrm{cm}$. Hence, the value of $a+b$ is 16 .