Geo geo

Let $\bold{a}$ represent the position vector of point $A$ , and so on. From the midpoint conditions on the sides: $\begin{aligned} 2\bold{m} &= \bold{a}+\bold{b} \\ 2\bold{p} &= \bold{b}+\bold{c} \\ 2\bold{n} &= \bold{c}+\bold{d} \\ 2\bold{q} &= \bold{d}+\bold{e} \end{aligned}$

Similarly, for $K$ and $L$ : $\begin{aligned} 4\bold{k} &= 2\bold{p}+2\bold{q} =\bold{b}+\bold{c}+\bold{d}+\bold{e} \\ 4\bold{l} &= 2\bold{m}+2\bold{n} =\bold{a}+\bold{b}+\bold{c}+\bold{d} \end{aligned}$

Subtracting, $4(\bold{k}-\bold{l})=\bold{e}-\bold{a}$

Hence $EA=|\bold{e}-\bold{a}|=4|\bold{k}-\bold{l}|=4KL=\boxed{100}$ .

Not only that, but the lines $KL$ and $EA$ are parallel.

Let the position coordinates of $A, B, C, D, E$ be $(0,0),(b, 0),(c_1,c_2),(d_1,d_2),(e_1,e_2)$ respectively. Then those of $M, N, P, Q, K, L$ are

$(\frac b2,0),(\frac{c_1+d_1}{2},\frac{c_2+d_2}{2}),(\frac{b+c_1}{2},\frac{c_2}{2}),(\frac{d_1+e_1}{2},\frac{d_2+e_2}{2}),(\frac{b+c_1+d_1+e_1}{4},\frac{c_2+d_2+e_2}{4}),(\frac{b+c_1+d_1}{4},\frac{c_2+d_2}{4})$ respectively.

Hence, $|\overline {KL}|=\dfrac {\sqrt {e_1^2+e_2^2}}{4}=25$

$\implies |\overline {EA}|=\sqrt {e_1^2+e_2^2}=\boxed {100}$ .