Find KN

Extend $BC$ and let the other end point be $E$ , where $\overline{BE}$ = 3.

Connect $\overline{EM}$ , $\overline{MK}$ , and $\overline{NL}$ . Let $\overline{NL} = a$ , $\overline{NK} = b$ .

$\overline{CL}$ = $\overline{LK}$ = $3 ... (1)$

$\overline{CN}$ = $\overline{NM}$ $... (2)$

$\overline{CK}$ = $\overline{KM}$ = $6 ... (3)$

From $(1)(2)$ we know that $\overline{MK} = 2a$

From $(2)(3)$ we know that $\overline{EM} = 2b$

According to Apollonius's Theorem:

$\overline{CA}^2$ + $\overline{CB}^2$ = 2( $\overline{CM}^2$ + $\overline{AM}^2$ )

$85 + 81 = 2(\overline{CM}^2 + 10 )$

$\overline{CM}$ = $\sqrt{73}$

$\overline{CN}$ = $\frac{\sqrt{73}}{2}$

$\overline{NK}^2$ + $\overline{NC}^2$ = 2( $\overline{NL}^2$ + $\overline{KL}^2$ )

$b^2 + \frac{73}{4} = 2a^2 + 18 \Rightarrow 2a^2 - b^2 = \frac{1}{4} ... (4)$

$\overline{EM}^2$ + $\overline{MK}^2$ = 2( $\overline{MB}^2$ + $\overline{BE}^2$ )

$4b^2 + 4a^2 = 38 ... (5)$

After doing $(5) - (4) \times 2$ we get:

$6b^2 = \frac{75}{2}$

$b^2 = \frac{25}{4} \Rightarrow b = \frac{5}{2} = \boxed{2.5}$

Assume B:(0,0), C:(9.0) and A:(a,b). Then, a²+b²=(2√10)² and (a-9)²+b²=(√85)² which makes A:(2,6),M:(1,3) & N:(5,3/2). This means KN² =(3-5)²+(0-3/2)²=25/4 or KN=5/2=2.5. As simple as that!

From the picture, we have $\cos\alpha=\frac{(2\sqrt{10})^2+9^2-(\sqrt{85})^2}{2\cdot9\cdot2\sqrt{10}}=\frac1{\sqrt{10}}$

Then, $\begin{aligned}MC^2&=9^2+(\sqrt{10})^2-2\cdot9\cdot\sqrt{10}\cos\alpha\\&=91-18\sqrt{10}\frac1{\sqrt{10}}\\&=91-18\\&=73\end{aligned}$

Half the length, $NC=\dfrac12MC=\dfrac{\sqrt{73}}{2}$ , so $NC^2=\dfrac{73}{4}$ .

Finding expression for $\beta$ : $\begin{aligned}\cos\beta&=\frac{9^2+(\sqrt{73})^2-(\sqrt{10})^2}{2\cdot9\cdot\sqrt{73}}\\&=\frac{144}{18\sqrt{73}}\\&=\frac{8}{\sqrt{73}}\\KN^2&=KC^2+NC^2-2KC\cdot NC\cdot\cos\beta\\&=36+\dfrac{73}{4}-2\cdot6\cdot\frac{\sqrt{73}}{2}\cdot\frac{8}{\sqrt{73}}\\&=36+\frac{73}{4}-48\\&=\frac{73}{4}-12\\&=\frac{25}{4}\\\therefore KN&=\boxed{\dfrac{5}{2}}\end{aligned}$

Use cos rule for ΔABC to find <A. cos A = (40+85-81)/(4√850) = 11/√850 , A=67.8. Use sin rule to find <C, 9/sin⁡A = (2√10)/sinC , C = 40.6 Use sin rule for ΔAMC to find NC=x. 4x^2 = 95 - 2√850 . 11/√850 , x = 4.272. Use cos rule for ΔKNC to find KN. KN^2 = 36 + (73/4) cos 20.3. KN=2.48