What is the length?

$\text{area of each rectangle}=4\times 1=4$ . Since there are $16$ congruent rectangles the total area of the figure $4\times 16=64$

It is given that $AP$ divides the area of the octagon into two equal parts. Hence, we have $[APFGH]=64/2=32$

$[APFGH]=[FGHQ]+[APQ]$

We know that $[FGHQ]=4\times 2=8$ $\color{#69047E}{\text{because, the rectangles are congruent}}$

$\therefore [APQ]=[APFGH] - [FGHQ]=32-8=24$

We have $\angle PQA =90^\circ$ $\color{#EC7300}{\text{because all are rectangles}}$

Hence, $\begin{aligned}\\& [APQ]=\dfrac{1}{2}\times AQ\times PQ=24 \\& \implies \dfrac{1}{2}\times 8\times PQ=24 ~~~~~~~~~~~~~~~~~~ \color{#3D99F6}{\text{because}} ~ \color{#333333}{AQ} ~ \color{#3D99F6}{\text{consists of widths of 8 congruent rectangles}}\\& \implies PQ=6\end{aligned}$

Hence, by Pythagorean theorem on $\triangle APQ$ we get

$\begin{aligned} AP=\sqrt{AQ^2+PQ^2} \\& =\sqrt{8^2+6^2}\\& =\sqrt{64+36}\\& =\sqrt{100}\\& =\boxed{10}\end{aligned}$

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Note: $\left[\cdot\right]$ represents the area of a polygon mentioned inside it

The lower half of the total area = 32 and consist of a right triangle + 2 (1x4) rectangles. Therefore 32 = (1/2)(b)(h) + 8.Since b = 8, transposing the 8, and multiplying by 2, 48 = 8h, so h = 6, and we have a 6,8,10 right triangle, so the length is 10. Ed Gray